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P 9
Normal Approximation
and Asymptotic
Expansions
b c.4

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Rabi N. Bhattacharya and R. Ranga Rao, Normal Approximation and Asymptotic Expansions

Normal Approximation
and Asymptotic
Expansions
ci
Rabi N. Bhattacharya
University of Arizona
Tucson, Arizona
R. Ranga Rao
University of Illinois at Urbana-Champaign
Urbana, Illinois
Society for Industrial and Applied Mathematics
Philadelphia

Copyright ? 2010 by the Society for Industrial and Applied Mathematics
This SIAM edition is an updated republication of the work first published by Robert
E. Krieger Publishing Company, Inc., in 1986, which was an updated and corrected
version of the original edition that was published by Wiley in 1976.
10987654321
All rights reserved. Printed in the United States of America. No part of this book
may be reproduced, stored, or transmitted in any manner without the written
permission of the publisher. For information, write to the Society for Industrial and
Applied Mathematics, 3600 Market Street, 6th Floor, Philadelphia, PA 19104-2688
USA.
Library of Congress Cataloging-in-Publication Data
Bhattacharya, R. N. (Rabindra Nath), 1937-
Normal approximation and asymptotic expansions / Rabi N. Bhattacharya, R. Ranga
Rao.
p. cm. -- (Classics in applied mathematics ; 64)
"Updated republication of the work first published by Robert E. Krieger Publishing
Company, Inc., in 1986"--Copr. p.
Includes bibliographical references and index.
ISBN 978-0-898718-97-3 (pbk.)
1. Central limit theorem. 2. Convergence. 3. Asymptotic expansions. I. Ranga
Rao, R. (Ramaswamy), 1935- II. Title.
QA273.67.B48 2010
519.2--dc22
2010031917
S.LaJTL is a registered trademark.

To owri and hantha

Contents
PREFACE TO THE CLASSICS EDITION xiii
PREFACE xv
LIST OF SYMBOLS xvii
CHAPTER 1. WEAK CONVERGENCE OF PROBABILITY
MEASURES AND UNIFORMITY CLASSES I
1.Weak Convergence, 2
2.Uniformity Classes, 6
3.Inequalities for Integrals over
Convex Shells, 23
Notes, 38
CHAPTER 2. FOURIER TRANSFORMS AND
EXPANSIONS OF CHARACTERISTIC
FUNCTIONS 39
4.The Fourier Transform, 39
5.The Fourier?Stieltjes Transform, 42
6.Moments, Cumulants, and Normal
Distribution, 44
7.The Polynomials Ps and the Signed
Measures Ps, 51
8.Approximation of Characteristic Functions
of Normalized Sums of Independent
Random Vectors, 57
9.Asymptotic Expansions of Derivatives of
Characteristic Functions, 68
10.A Class of Kernels, 83
Notes, 88
ix

X Contents
CHAPTER 3. BOUNDS FOR ERRORS OF NORMAL
APPROXIMATION 90
11.Smoothing Inequalities, 92
12.Berry?Esseen Theorem, 99
13.Rates of Convergence Assuming Finite
Fourth Moments, 110
14.Truncation, 120
15.Main Theorems, 143
16.Normalization, 160
17.Some Applications, 164
18.Rates of Convergence under Finiteness of
Second Moments, 180
Notes, 185
CHAPTER 4. ASYMPTOTIC EXPANSIONS-
NONLATTICE DISTRIBUTIONS 188
19.Local Limit Theorems and Asymptotic
Expansions for Densities, 189
20.Asymptotic Expansions under Cramer's
Condition, 207
Notes, 221
CHAPTER 5. ASYMPTOTIC EXPANSIONS?LATTICE
DISTRIBUTIONS 223
21.Lattice Distributions, 223
22.Local Expansions, 230
23.Asymptotic Expansions of Distribution
Functions, 237
Notes, 241
CHAPTER 6. TWO RECENT IMPROVEMENTS 243
24.Another Smoothing Inequality, 243
25. Asymptotic Expansions of
Smooth Functions of
Normalized Sums, 255

Contents xi
CHAPTER 7. AN APPLICATION OF STEIN'S METHOD 260
26. An Exposition of Gotze's Estimation of the Rate
of Convergence in the Multivariate Central Limit
Theorem, 260
APPENDIX A.I. RANDOM VECTORS AND
INDEPENDENCE 285
APPENDIX A.2. FUNCTIONS OF BOUNDED VARIATION
AND DISTRIBUTION FUNCTIONS 286
APPENDIX A.3. ABSOLUTELY CONTINUOUS.
SINGULAR, AND DISCRETE
PROBABILITY MEASURES 294
APPENDIX A.4. THE EULER?MACLAURIN SUM
FORMULA FOR FUNCTIONS OF
SEVERAL VARIABLES 296
REFERENCES 309
INDEX 315

Preface to the Classics Edition
It is with great pleasure that the authors welcome the publication by SIAM
of the edited reprint of Normal Approximation and Asymptotic Expansions. The
original edition was published in 1976 by Wiley, followed by a Russian transla-
tion in 1982 and an edited version with a new chapter by Krieger in 1986.
The book has been out of print for nearly twenty years. Statistical applications
such as "higher order" comparisons of efficiency, and the evaluation of the
improvement over the classical central limit theorem due to the widely popular
and important bootstrap methodology of Efron, have led to a renewed interest
in the subject matter of the book. We also note with a measure of happiness
that the theory of asymptotic expansions for sums of weakly dependent random
variables/vectors due to Gotze and Hipp made use of some of the formalism
and estimation in our book. We have controlled an initial impulse to present
this theory, as it would make the book substantially increase in size and take us
much time to ready it for publication.
Keeping to independence, however, a short new chapter is added on
Gotze's application to the multivariate CLT of an ingenious method of Stein.
The exposition, and a somewhat modified treatment presented here of the
rather difficult original paper, resulted from a collaboration between one
of the authors and Professor Susan Holmes.
Finally, we are deeply appreciative that our colleague Professor William
Faris has always liked the book. His support, as well as that of SIAM
acquisitions editor Sara Murphy, made the publication of the present reprint
possible. We are indebted to Bill and Sara.
xiii

Preface
This monograph presents in a unified way various refinements of the
classical central limit theorem for independent random vectors and in-
cludes recent research on the subject. Most of the multidimensional results
in this area are fairly recent, and significant advances over the last 15 years
have led to a fresh outlook. The increasing demands of application (e.g., to
the large sample theory of statistics) indicate that the present generality is
useful. It is rather fortunate that in our context precision and generality go
hand in hand.
Apart from some material that most students in probability and statistics
encounter during the first year of their graduate studies, this book is
essentially self-contained. It is unavoidable that lengthy computations
frequently appear in the text. We hope that in addition to making it easier
for someone to check the veracity of a particular result of interest, the
detailed computations will also be helpful in estimations of constants that
appear in various error bounds in the text. To facilitate comprehension
each chapter begins with a brief indication of the nature of the problem
treated and its solution. Notes at the end of each chapter provide some
history and references and, occasionally, additional facts. There is also
an Appendix devoted partly to some elementary notions in probability
and partly to some auxiliary results used in the book.
We have not discussed many topics closely related to the subject matter
(not to mention applications). Some of these topics are "large deviation,"
extension of the results of this monograph to the dependence case, and
rates of convergence for the invariance principle. It would take another
book of comparable size to cover these topics adequately.
We take this opportunity to thank Professors Raghu Raj Bahadur and
Patrick Billingsley for encouraging us to write this book and giving us
xv

xvi Preface
advice. We owe a special debt of gratitude to Professor Billingsley for his
many critical remarks, suggestions, and other help. We thank Professor
John L. Denny for graciously reviewing the manuscript and pointing out a
number of errors. We gratefully acknowledge partial support from the
National Science Foundation (Grant. No. MPS 75-07549). Miss Kanda
Kunze and Mrs. Sarah Oordt, who did an excellent job of typing the
manuscript, have our sincere appreciation.
R. N. BHATTACHARYA
R. RANGA RAO
Note
In this reprinted edition a new chapter (Chapter 6) has been added and misprints in the original
edition corrected.

List of Symbols
A\B set of all elements of A not in B: (1.4)
A+y (x+y:xEA): (5.5)
A ` set of all points at distances less than e
from A : (1.17)
A -` set of all x such that the open ball of
radius a centered at x is contained in A:
(2.38)
9 a generic class of Borel sets
c a* (d : ?), special classes of Borel Sets:
d. (d : 41o, v) (17.3), (17.52)
a„ (14.64)
a, /3 usually nonnegative vectors with integral
coordinates; sometimes positive numbers
^aj sum of coordinates of a nonnegative
integral vector
B a generic Borel set
B, B„ positive square roots of the inverses of
matrices V, V,,: (9.7), (19.28)
B (x : e) open ball of radius a centered at x:
(1.10)
n1'` Borel sigma-field of R k
Cl(A) closure of A
C class of all convex Borel subsets of R k
c(B) convex hull of B: Section 3
xvti

xviii List of Symbols
cov(X, Y) covariance between random variables
X, Y: (A.1.5)
Cov(X) matrix of covariances between co-
ordinates of a random vector X:
Appendix A. 1
D average covariance matrix of centered
truncated random vectors Z,,...,Z,,:
(1-4.5)
D° ath derivative: (4.3)
d (0, aA) euclidean distance between the origin
and aA: Section 17
d0(G1, G2) (17.50)
dp Prokhorov distance: (1.16)
dBL bounded Lipschitzian distance: (2.51)
d(P,4) (12.47)
Det V, Det D determinant of a matrix V or D
det L absolute value of the determinant of the
matrix of basis vectors of a lattice L:
(21.20)
0(A, B) Hausdorff distance between sets A and
B: (2.62)
(14.4)
0,,,(e) (14.105), (14.106)
Ors (15.7)
^^ J (17.55)
S,!s (18.4)
aA topological boundary of A: (1.15)
EX, E(X) expectation or mean of a random
variable or random vector X:
(A. 1.2), (A. 1.3)
E, E generic small numbers
E symbol for "belongs to"
f
Fouriertransform of a function f: (4.5)
f (4.4)
f(x) f(x+y): (11.5)
f*g convolution of functions f and g: (4.9)
f*" n-fold convolution of a function f: (4.11)

List of Symbols xix
a generic class of Borel-measurable
functions
fundamental domain for the dual lattice
L*: (21.22)
normal distribution on R k with zero
mean and identity covariance matrix
density of 1
(Dm. v
normal distribution with mean m and
covariance V
m
V density of 4,,, v: (6.31)
(D,o (15.5), (18.10)
G.,m, g0, a special probability measure and its
density: (10.7)
gT
(16.7)
Y(f:E), Y*(f:E) (11.8), (11.18)
11i ^ (9.8), (19.32)
I k X k identity matrix
IA indicator function of the set A
Int(A) interior of A
K( a smooth kernel probability measure
assigning either all or more than half
its mass to the sphere B (x : e):
(11.6), (11.16), (15.26)
X„ with cumulant, average of with cumulants
of X,,...,X,,: (6.9), (9.6), (14.1)
average of with cumulants of centered
truncated random vectors Z,, ... , Z,,:
(14.3)
X,,,j, X,,.,, with cumulant of X^, I < j < n, and their
average: (9.6), Sections 19, 20
X,(z) (6.16)
L a lattice: Section 21
L* lattice of periods of
f, f being the char-
acteristic function of a lattice random
vector: (21.9), (21.19)
L(c,d) a Lipschitzian class of functions: (2.50)
11,11 Liapounov coefficient: (8.10)
A, A smallest and largest eigenvalues of an
average covariance matrix V:
Section 16

xx List of Symbols
Ak Lebesgue measure on Rk
A,.,,(F) (23.8)
M,(f), MO(f) (15.4)
M1(x : e), mj(x : E) supremum and infimum off in B (x : e):
(11.2)
)i, set of all finite signed measures on a
metric space.
?+, ?-, ^ ?I positive, negative, and total variations of
a finite signed measure ?: (1.1)
Il ?dl variation norm of a signed measure ?:
(1.5)
? Fourier-Stieltjes transform of ?: (5.2)
?• v convolution of two finite signed measures
?, v: (5.4)
?•" n-fold convolution of ?: (5.6)
?o T -' signed measure induced by the map T:
(5.7)
?Q ath moment, average of ath moments of
X1,...,X,,: (6.1), (14.1)
average of ath moments of centered
truncated random vectors Z,,...,Z,,:
(14.3)
?.(t), f3,(t) (8.4)
v! PI!v2!...Pk! where v=(P,,••.,Pk) is a non-
negative integral vector
v,, vo special signed measures: (15.5)
P a probability measure, a polyhedron
set of all probability measures on a
metric space
P characteristic function of a probability
measure P: (5.2)
P,(z : (x,)) a special polynomial in z: (7.3)
P.(-4
0,v: {x,}) a polynomial multiple of 0o.v: (7.11)
P,(-'o, v : {x,}) signed measure whose density is
P,(-$o,v: (X„})
Pa a special polyhedron: (3.19)
point masses of normalized lattice
random vectors X1,.. .,X17: (22.3)
P;,(Y.,,,)
point masses of normalized truncated lat-
tice random vectors: (22.3)

List of Symbols xxi
Q„ distribution of n -'/Z(XI + • • •
where X1,...,X,, are independent random
vectors having zero means and average
covariance matrix V (or 1)
Q„ distribution of n -1 2(Y, + + Y„),
where YT's are truncations of XD's:
(14.2)
Q„ distribution of n -1 '2(Z1 + + Z„),
where Z^=Y^?EYE: (14.2)
9n,m, 9n,m local expansions of point masses of Q,
Q,,' in the lattice case: (22.3), (22.38),
(23.2)
p(x, A) distance between a point x and a set A:
(1.18)
p, sth absolute moment, average of sth
absolute moments of X1,.. .,X:
(6.2), (9.6), (14.1)
p; average of sth absolute moments of
centered truncated tandom vectors
(14.3)
p absolute moment of X^, I < j < n, and
their average: (14.1), (17.55)
SS, S. special periodic functions:
(A.4.2), (A.4.14)
S Schwartz space: (A.4.13)
ak_I surface area measure on the unit sphere
of R
k:
Section 3
11Th norm of a matrix T: (14.17)
T
r (16.6)
T(f : 2E), T*(f : 2E) (11.8), (11.18)
V average of covariance matrices of ran-
dom vectors X1,...,X: (9.6), (14.5)
wf(A) oscillation off on A: (2.7), (11.1)
wf(x : E) oscillation off on B (x : E): (2.7), (11.3)
Zj(E : ?) average modulus of oscillation off with
respect to a measure ?: (11.23)
sup 1(: s): (11.24)
y
IxI x,I+ • • • +IxkI, where x=(xl,...,xk):
(4.8)

xxii List of Symbols
ya,n+ ya,n

(22.3)
z+

set of all nonnegative integers
(Z+)k

set of all k-tuples of nonnegative integers
il•Ii, <,>

euclidean norm and inner product
II • IIP

Lp-norm
z

set of all integers

CHAPTER 1
Weak Convergence of Probability
Measures and Uniformity Classes
Let Q be a probability measure on a separable metric space S every open
ball of which is connected (e.g., S=R"). In the present chapter we
characterize classes f of bounded Borel-measurable functions such that
sup I f fdQ„- f fdQI-0 (n--*oo), (1)
Jeff S S
for every sequence { Q„ : n> I) of probability measures converging weakly
to Q. Such a class is called a Q-uniformity class. It turns out that is a Q
uniformity class if and only if
sup w1(S)<oc, urn f sup f wj(x:e)Q(dx)l =0, (2)
JEc 4° L JET-, S J
where wJ(S) is the (total) oscillation off on S. and w1(x : e) its oscillation
on the open ball of radius a centered at x. This suggests that the ap-
propriate characteristics of f on which the rate of convergence f fdQ„
-. f fdQ depends are (i) w1(S) and (ii) the average oscillation function
€.- fwJ(x : e)Q (dx). Specialized to indicator functions of Borel sets A, this
says that the rate of convergence Q„(A)-►Q(A) depends on the function
E_*Q((8A)`), where aA is the boundary of A and (8A) is the set of all
points whose distances from aA are less than E. We have pursued this line
of thinking in Chapters 3 and 4 to obtain appropriate rates of convergence
for the central limit theorem.
Section 1 contains a brief review of those aspects of weak conver-
gence theory that are relevant for proving results on characterization of

Weak Convergence and Uniformity Classes
uniformity classes in Section 2. These two sections are not used in the
sequel (except as motivation). In Section 3 we obtain estimates such as
sup 4((aC))<d(k)e (e>0), (3)
CEC
where 4) is the standard normal distribution on R', C is the class of all
(Borel-measurable) convex subsets of R k, and d(k) is a positive number
depending only on k. We have several occasions in Chapters 3 and 4 to use
these estimates for deriving rates of convergence Q„(C)-->4'(C), C E C',
where Q„ is the distribution of the normalized sum of n independent
random vectors.
1. WEAK CONVERGENCE
In this section we briefly review some standard results in the theory of
weak convergence of probability measures.
Throughout this section S denotes a metric space with a metric p. The
Borel sigma field 3 of S is the smallest sigma-field containing the class of
all open subsets of S. We say is a (signed) measure on S if it is a (signed)
measure defined on . The class of all finite signed measures on S is
denoted by OiL, and the subclass of O1i, comprising all probability mea-
sures is denoted by 'P. Given a finite signed measure s on S, one defines
three associated set functions called the positive, negative, and
total variations of ?, respectively, by
(B)=sup{?(A):ACB, AE),
(B)=?-(B)=?inf( (A):ACB, AE°.1J), (BEAU) (1.1)
1?1=?++
tL .
The so-called Jordan-Hahn decompositions asserts that ?+ and s- (and,
therefore I s) are finite measures on S satisfying
(1.2)
For every finite signed measure s on a separable metric space S, we define
the support of as the smallest closed subset of S whose complement has
I ?I-measure zero: that is,
support of ?= n {F:Fclosed, I?I(S\F)=0}, (1.3)
tSee Halmos [1], pp. 121-123.

Weak Convergence 3
where for any two sets A, B we write
A\B={x:xEA,x^B}. (1.4)
Note that the separability of the metric space S ensures that the comple-
ment of the right side of (1.3) has zero I SI-measure.
The class OIL of (set) functions on J into R 1 is a real linear space with
respect to pointwise addition and multiplication by real scalars. It is a
Banach space when endowed with the variation norm
II?II=IIl(S) (ttEOIL). (1.5)
Let C(S) denote the class of all complex-valued, bounded, continuous
functions on S. The weak totopogy on Olt, is the weakest topology (on OIL)
that makes the maps
?? f fd? [f EC(S)] (1.6)
on cX into the complex field C continuous. The right side of (1.6) always
stands for the Lebesgue integral of (a ?-integrable, complex-valued, Borel-
measurable function) f on S. The Lebesgue integral of f on a Bore! set B is
denoted by
f
Bfd?. (1.7)
When it becomes necessary to indicate the variable of integration, we also
write
f f (x) ?(dx) (1.8)
instead of f f d?.
In this monograph we are particularly concerned with the relativized
weak topology on the class 6P of all probability measures on S. In this
topology convergence of a sequence (Q„) of probability measures to a
probability measure Q means
lira f fdQ„= f fdQ (1.9)
for every f in C(S). The following theorem gives several characterizations
of weak convergence of a sequence of probability measures.

4 Weak Convergence and Uniformity Classes
THEOREM 1.1. Let S be a metric space. Let Q. (n= 1,2,...), Q be
probability measures on S. The following statements are equivalent.
(i)Q„ converges weakly to Q.
(ii)linen f f dQ,, = ffdQdQ for every uniformly continuous f in C (S).
(iii)urn Q. (F) < Q (F) for every closed subset F of S.
(iv)lint Q„(G)> Q(G) for every open subset G of S.
n
For a proof of this theorem we refer to Billingsley [1] (Theorem 2.1, pp.
11-14) or Parthasarathy [1] (Theorem 6.1, pp. 40-42).
Let B (x : E) denote the open ball with center x and radius E,
B(x:E)={y:yES, p(x,y)<E} (xES, c>0). (1.10)
For an arbitrary real-valued function f on S we define, for each positive
number e, the oscillation function wj(-

wf(x:E)= sup {I f(z)-f(y)I:y,zEB(x:c)} (x ES). (1.11)
For a complex-valued function f= g + ih (g, h real-valued), define
Wf(X:E)?wg(X:E)+Wh(X:E) (xES, E >0). (1.12)
The oscillation function is Borel-measurable on the (Borel-measurable) set
on which it is finite.t The set of points of discontinuity of f is Borel-
measurable and can be expressed as
(x:w j(x: 1)+0 as n-moo}. (1.13)
n 1
Now let Q be a probability measure on S. A complex-valued function f on
S is said to be Q-continuous if its points of discontinuity comprise a set of
Q-measure zero. In particular, if the indicator function IA of a set A, taking
values one on A and zero on the complement of A, is Q-continuous, we say
A is a Q-continuity set. Since the set of points of discontinuity of IA is
precisely the boundary 8A of A, A is a Q-continuity set if and only if
Q(aA)=0. (1.14)
Recall that the (topological) boundary aA of a set A is defined by
aA =Cl(A)\Int(A), (1.15)
tSee relations (11.1)?{11.4) and the discussion following them.

Weak Convergence 5
where C1(A), Int(A) are the closure and interior of A, respectively.
LEMMA 1.2. Let Q be a probability measure and f a complex-valued,
bounded, Borel-measurable function on a metric space S. The following
statements are equivalent.
(i)f is Q-continuous.
(ii)lim Q((x:wf(x:e)>6))=0 for every positive S.
(iii) urn Jwf(x : e) Q (dx) = 0.
Proof Let D be the set of discontinuities off. As 40 the sets { x : wf(x : e)
>6) decrease to a set Da. Now (i) means Q(D)=O and (ii) means
Q(D8)=0 for all 6>0. Since D= U Did, (i) and (ii) are equivalent.
n>I
Since, as 40, the functions wl( : e) are uniformly bounded and decrease to
a function that is strictly positive on D and zero outside, (iii) is equivalent
to Q(D)=0. Q.E.D.
The next theorem provides two further characterizations of weak con-
vergence of a sequence of probability measures.
THEOREM 1.3. Let Q„ (n = 1, 2, ... ), Q be probability measures on a
metric space S. The following statements are equivalent.
(i){ Q„ } converges weakly to Q.
(ii)linm Q„(A)=Q(A) for every Borel set A that is a Q-continuity set.
(iii) lim Jf dQ„ = Jf dQ for every complex-valued, bounded, Borel-
measurable Q-continuous function f.
Although it is not difficult to prove this theorem directly, we note that it
follows as a very special case of Theorem 2.5.
We conclude this section by recalling that if the metric space S is
separable, then the weak topology on 9l is metrizable and separable, and
that a metrization is provided by the Prokhorov distance dp between two
probability measures Q, and Q2, which is defined by
a,(Q1,Q2)=inf {E:c>O, Q1(A)<Q2(A`)+e
and Q2(A) < QI(A`)+a for all Borel sets A ), (1.16)
where
A`={x:xES, p(x,A)<e}. (1.17)

6 Weak Convergence and Uniformity Classes
and
p(x,A)=inf (p(x,y):yEA}. (1.18)
For these and other details we refer the reader to Billingsley [1], (Appendix
III, pp. 233-242) or Parthasarathy [1] (Chapter II, pp. 39-55).
2. UNIFORMITY CLASSES
Unless otherwise specified, throughout this section S denotes a separable
metric space.
Let Q be a given probability measure on S. Our main task in this section
is to characterize those classes f of complex-valued, bounded, Borel-
measurable functions on S for which
lim sup f fdQ„? f fdQI =0 (2.1)
n fEc
for every sequence of probability measures (Q„) converging weakly to Q.
Such a class is called a Q-uniformity class of functions. A class 9 of
Borel subsets of S is called a Q-uniformity class of sets if
lim sup IQ(A)?Q(A)I=0 (2.2)
n AE(^
for every sequence of probability measures {Q„) converging weakly to Q.
Thus (t' is a Q-uniformity class of sets if and only if = { IA : A E Q) is a
Q-uniformity class of functions. We need some preparation before we can
characterize uniformity classes.
In the following lemma we deal with (signed) measures on a sigma-field
of subsets of an abstract space I. For a finite signed measure on this
space the variation norm II ?II is again defined by (1.5) with 2 replacing S
(and C replacing i3( ).
LEMMA 2.1. (Scheffe's theorem) Let (St, LS ,A) be a measure space. Let
Q„ (n = 1,2,...), Q be probability measures on (S2, in ) that are absolutely
continuous with respect to A and have densities (i.e., Radon-Nikodym deriva-
tives) q„(n = 1,2,...), q, respectively, with respect to A. If (q„) converges to q
almost everywhere (A), then
linmIIQ,,?Q11=O.

Uniformity Classes 7
Proof Let h„ = q ? q. Clearly
f
h„dA=0 (n=1,2,...),
so that
IIQ,?QII= f h„da ? f h„dX =2 f h„dx =2 f h„•!{ti,>o)dX. (2.3)
(h„>0) (h„ G0) {h„>0)
The last integrand in (2.3) is nonnegative and bounded above by q. Since it
converges to zero almost everywhere, its integral converges to zero. Q.E.D.
LEMMA 2.2. Let S be a separable metric space, and let Q be a probability
measure on S. For every positive a there exists a countable family (Ak : k
= 1,2,...) of pairwise disjoint Borel subsets of S such that (i) U (Ak: k
= 1,2,...) = S, (ii) the diameter of Ak is less than a for every k, and (iii)
every Ak is a Q-continuity set.
Proof. For each x in S there are uncountably many balls (B (x : 8) : 0< 6
<e/2) (perhaps not all distinct). Since
aB (x : 6) C { y : p(x,y) = S }, (2.4)
the collection { aB (x : 3) : 0 < </2) is pairwise disjoint. But given any
pairwise disjoint collection of Borel sets, those with positive Q-probability
form a countable subcollection. Hence there exists a ball B (x) in
{ B (x : S) : 0 < S < e/2) whose boundary has zero Q-probability. The col-
lection (B (x) : x E S) is an open cover of S and, by separability, admits a
countable subcover { B (xk) : k = 1, 2, ... ), say. Now define
Ai=B(x1), A2=B(x2)\B(x1),..., Ak=B(xk)(U_i'B(x1)),....
(2.5)
Clearly each Ak has diameter less than e. Since
aA =a(S\A), a(A n B)C(aA)U(aB), (2.6)
for arbitrary subsets A, B of S. it follows that each Ak is a Q-continuity set.
Q.E.D.
To state the next lemma, we define, in addition to wf(• : e), the oscillation
w^(A) of a complex-valued function f on a set A by
wf(A) =sup {If(x)?f(y)I:x,yEA} (ACS). (2.7)

8 Weak Convergence and Uniformity Classes
Clearly,
w1(x:E)=w1(B(x:E)) (xES, E>O). (2.8)
LEMMA 2.3. Let f be a complex-valued, bounded, Bore/-measurable func-
tion on a separable metric space S. For each pair of positive numbers E and 8
there exists a countable collection of pairwise disjoint Bore! sets (Nk : k
= 1,2,...) such that (i) U (Nk : k = 1,2,...) D (x : w1(x : E) > S ), (ii) the diam-
eter of Nk is less than 6E for every k, and (iii) wr(Nk)> S for every k.
Proof. Let G={x:wf(x:c)>S). Let (y:n=1,2,...) be dense in G, so
that G C U (B (y E) : n =1,2,...). Let x, =y1; let x2 be the first y„ whose
n
distance from x, is at least 2E; let x3 be the first y„ beyond x2 whose
distance from (x1,x2) is at least 2E, and continue to get a countable set
{ x„ : n = 1,2,...) such that (B (xn : E) : n = 1,2,...) is a pairwise disjoint
collection. Also, since each y„ is within a distance of 2E from some xx,
GCU{B(x,,:3E):n=1,2,...).
Let B=U{B(x,,:c):n=1,2,...) and define
Mk=B(xk:3E)(B U U{B(xk,:3E):1 <k'<k)),
Nk=B(xk:E)UMk.
Note that the Mk's are pairwise disjoint and U { Mk : k =1, 2, ...) J G B;
also, Mk is disjoint from B. Hence (Nk : k = 1,2,...) is a pairwise disjoint
collection of sets whose union contains G. Since Xk E G and B (xk : E) c Nk,
it follows that wf(Nk) > wf(xk : E)> 8. Finally, Nk C B (xk : 3E), so that the
diameter of Nk is not more than 6E. Q.E.D.
We are now ready to prove the main theorem of this section.
THEOREM 2.4. Let S be a separable metric space and Q a probability
measure on it. A family F of complex-valued, bounded, Borel-measurable
functions on S is a Q- uniformityclass if and only if
(i) sup w1(S) < oo,
In
(ii)lim sup (Q ({ x : wf(x : E)>S))) = 0 for every positive 8. (2.9)
fEJ
Proof. The theorem is proved, without any essential loss of generality, for
a class S of real-valued, bounded, Borel-measurable functions.

Uniformity Classes 9
SUFFICIENCY. Assume that (2.9) holds. Let c, a be two positive numbers.
Let (Ak : k = 1,2,...) be a partition of S by Borel-measurable Q-continuity
sets each of diameter less than e. Lemma 2.2 makes such a choice possible.
Define the class of functions '+< , by
f =1.
Y.ckIA,:Ick
lccforallk}. (2.10)
k
Then is a Q-uniformity class. To prove this, suppose that {Q„} is a
sequence of probability measures converging weakly to Q. Define func-
tions q,, (n= 1,2,...), q on the set of all positive integers by
q.(k)=Q,,(Ak),
q(k) = Q
(Ak), (k=1,2,...). (2.11)
The functions q, q are densities (i.e., Radon?Nikodym derivatives) of
probability measures on the set of all positive integers (endowed with the
sigma-field of all subsets) with respect to the counting measure that assigns
mass one to each singleton. Since each Ak is a Q-continuity set, {q(k)}
converges to q(k) for every k. Hence, by Scheffe's theorem (Lemma 2.1),
lim 2 Iq(k)?q(k)I=0. (2.12)
k
Therefore
sup
(If fdQ^?f fdQl •f E `'`J
=sup {
t
12 ckq„(k)? 2 ckq(k) : Ickl < c for all k }
k k JJJ
^ c2Iq„(k)?q(k)I-0 as n?oo. (2.13)
k
Thus we have shown that 9 ,, is a Q-uniformity class. We now assume,
without loss of generality, that every function f in F is centered; that is,
inf f (x) = ? sup f (x), (2.14)
xES
xES
by subtracting from each f the midpoint cf of its range. This is permissible
because
sup IJ fd(Q,,?Q)I = sup
If (f?cf)d(Q„?Q)I (2.15)
f EN f r=

10 Weak Convergence and Uniformity Classes
whatever the probability measure Q. For each f E define
gJ(x) XEAkf(x)
for xEA k (k=1,2,...),
h1(x)= sup f(x) for xEAk (k=1,2,...). (2.16)
xEAk
Note that g,h E 9 ^,
gJ< f<hj, hf(x)?gf(x)=wf(Ak) for xEAk (k=1,2,...).
(2.17)
It follows that
f gjd(Q,?Q)? f (hJ?gJ)dQ= f gjdQ.? f hjdQ
<f.fdQn ? f fdQ<fhjdQ„ ? f gjdQ
= f h1d(Q,?Q)+ f (hJ?gj)dQ,
(2.18)
so that
sup Iffd(Q„?Q)I< sup Iffd(Qn?Q)I+ sup f(hJ?gj)dQ.
JE f J'E!^.. JE v
(2.19)
Since ^ is a Q-uniformity class,
urn sup Iffd(Q.?C))I< sup f (hJ?gj)dQ
n JEc JEeq
= Sup I wf (Ak)Q (Ak)
JE'F k
6 sup c Q(Ak)+ö
JE J [ (wj(A.)>8 )
<csupQ(tx:(Jj(x:E)>S))+s, (2.20)
JE J
since the diameter of each Ak is less than e. First let 40 and then let 8j,0.
This proves sufficiency.

Uniformity Classes 11
NECESSITY. Assume (2.9i) does not hold. Then there exists a sequence
{ f,,) c such that
mJ (S)>n (n=1,2,...). (2.21)
Thus if we write
a„=inf(f„(x):xES}, Q=sup(f„(x):xES}, (2.22)
then /3„ ? a,, > n for all n. Divide the closed interval [a,,, /3J into n disjoint
subintervals of equal length. There exists a subinterval I, such that
Q(f^ I(I.,))>n
(n=1,2,...). (2.23)
Also, since /3,, ? an > n, there exists a point x,, in S for which
If„(x,,)?tl. n2 for all tECI(I,,) (n=1,2,...). (2.24)
Now define a probability measure Q„ by adding a point mass 1/n at x,,
and subtracting this mass proportionately from subsets of f,^ '(I„); that is,
Q.(A)=Q(A
n '(In))+ ? (A)
+f 1? nQ(
f^ ^(I^)) ]Q(Anf„ '(I.)),
(AE), (2.25)
where Sx. is the probability measure degenerate at x„ (i.e., Sx. ({ x,, }) = 1).
Note that
IIQ.-Q11=n, (2.26)
so that Q,, converges in variation norm and, therefore, weakly to Q. But
f f dQ,,? Jf,,dQl = n II„(x,,)? Q f^
1^ (I^) 3J (i^)l,,dQ

12 Weak Convergence and Uniformity Classes
for some! in Cl(I„). Thus, by (2.26),
f fn dQ. ? f f" dQI >'
(2.28)
2n
for all n, implying that F is not a Q-uniformity class.
Next assume (2.9ii) does not hold. This means that there exist positive
numbers 8 and ri, a sequence {?) of positive reals converging to zero, and
a sequence (f) c 5 such that
Q({x:w1(x:f„)>6})>r1>0 (n=1,2,...). (2.29)
Let (Nk.,, : k = 1,2,...) be a countable collection of pairwise disjoint Borel
sets satisfying
(i)U(Nk,„:k=1,2,...)D(x:wJ(x:e„)>8),
(ii)diameter of Nk.A < 6E„ for each k,
(iii) wf (Nk.„)> 8 for each k (n=1,2,...).
Such a collection exists (for each n) by Lemma 2.3. Given n, for each k
choose two points xk,n, Yk,n in Nk „ such that
f,, (Y) ?f,, (x)> 8 (k=1,2,...). (2.30)
Thus
T, Q(Nk,n)f (Yk,n)? 2 Q(Nk,,,)fn(xk,n)>s7
1
,
k k
which implies that either
I f f.dQ?YQ(Nk.,,)fl(xk.n)>
(n=i,2,...), (2.31)
k Nk.4 k
or
7.Q(Nk.f)fn(Yk.f)-2 ffndQ> (n=1,2,...). (2.32)
k k Nk.,
If (2.31) holds, then define Q„ by
Q„(A)=Q(A\U (Nk..:k=1,2,... ))
+2Q(N k..)S.,..(A) (A E ); (2.33)
k

Uniformity Classes 13
if (2.32) holds, then define Q„ by
Q(A)=
Q(A\U (Nk.f:k=1,2,... ))
+2Q(Nk.,)8) (A)
(A E (2.34)
k
Suppose (2.31) holds. Let f be a uniformly continuous complex-valued
function on S. Then
f fdQn? f fdQl =l Y. f
Nk..
(f(xk.n)?f)dQl
k
wf(Nk.fl)Q(Nk.n)
k
<sup(wf(Nk,,,):k=1,2,... ). (2.35)
The right side of (2.35) goes to zero as n goes to infinity, since the diameter
of Nk.„ is, for all k, not more than 6e„ (which goes to zero). Hence {Q„)
converges weakly to Q by Theorem 1.1. On the other hand
f f.dQ ? f fndQn=2(f f.dQ ?Q(Nk.n)fn(xk.,,))>S2 (n=1,2,...),
k N,,,
by (2.31), which shows that 5 is not a Q-uniformity class. A similar
argument applies if (2.32) holds. Q.E.D.
Remark. Let S be a separable metric space, with Q, and Q2 two probability measures on it
such that Q2 is absolutely continuous with respect to Q1. It follows from Theorem A.3.1 (see
Appendix), which characterizes absolute continuity, that every Q,-uniformity class of func-
tions is also a Q2-uniformity class.
The following variant of Theorem 2.4 is also useful.
THEOREM 2.5. Let S be a separable metric space and Q a probability
measure on it. A family 9 of complex-valued, bounded, Bore/-measurable
functions on S is a
Q- uniformity class if and only if
(i)sup wf(S) < oo,
In
(ii)lim sup f w1(x : e)Q (dx) = 0. (2.36)
CIO JE
Proof. Suppose is a Q-uniformity class. By Theorem 2.4, (2.9) holds.
Let c =sup {wj(S) : f E ). Given a positive 6 there exists a positive

14 Weak Convergence and Uniformity Classes
number Eo(S) such that
sup Q I { x : wj(x : f) > ) / < 2c

(2.37)
JEg; `l
for every E less than Eo(S). Hence for all f in +
f wf(x:€)Q(dx)< f)
{X:wf(x:t)<
21
4S
whenever a is less than Eo(S). This proves necessity of (2.36).
Conversely, suppose (2.36) holds. Choose and fix a positive number S.
Given a positive, there exists a positive number E,(r^) such that
sup f wj(x:c)Q(dx)<6r1
fEgi
for all E less than E,(11). Hence for every f in f,
Q({X:W1(X:E)>S})< -k
f
Wf(x:E)Q(dx)<77
for all a less than E,(ij). Thus (2.9) holds. Q.E.D.
In order to specialize the above theorem to Q-uniformity classes of sets,
we define
A`= {x: p(x,A)<e} = U {B(x:e):xEA),
A-`=(x:B(x:E)CA}=S(S\A)`, (ACS, E>O). (2.38)
The set A ` was also defined earlier in (1.16). Note that A ` is open and A -`
is closed.
COROLLARY 2.6. A class C?' of Bore! subsets of a separable metric space
S is a Q-uniformity class if and only if
lim sup Q(A`\A -`)=0. (2.39)
110 A E a
If every open ball of S is connected, then
A`\A-`=(aA)` (ACS, E>0), (2.40)

Uniformity Classes 15
so that in this case is a Q-uniformity class if and only if
lim sup Q((aA)`)=0. (2.41)
fio AE&
Proof. For any arbitrary set A, wrA(x : e) is one if and only if B (x : e)
intersects both A and S\A; otherwise it has the value zero. Therefore
W]A(X:E)=,A'\A-.(x) (XES, E>0). (2.42)
Hence
f
wjA(x:e)Q(dx)=Q(A`\A -`) (ACS, e>0). (2.43)
Since w,A (S) < I for all sets A, it follows from Theorem 2.5 that (2.39) is a
necessary and sufficient condition for & to be a Q-uniformity class.
We now prove (2.40) under the hypothesis that every open ball of the
metric space S [separability is not needed for the validity of (2.40)] is
connected. First we show that the relation
(aA)`CA`\A -` (ACS, e>0) (2.44)
is valid in every metric space S. For suppose xE(aA)`. Then there exist a
positive c' smaller than and a pointy in aA such that p(x,y) < e'. Since y
is a boundary point of A, there exist two points z, and z2, with z, in A and
z2 in S \A, such that p(y, z.) < E ? e' for i = 1, 2. Thus p(x, z.) < p(x,y) +
p(y, z;) < e for i= 1, 2. This means that x E A ` and x A - `, which proves
(2.44). Next we assume that every open ball of S is connected. If x E A `\
A -`, then
AnB(x:e)#4, (S\A)nB(x:e)^. (2.45)
We now suppose that xE(8A) (and derive a contradition). This means
B(x':e)n 8A=0, (2.46)
so that
B(x:e)=((S\Cl(A))n B(x:E))u(Int(A)nB(x:e)), (2.47)
since S = (S \C 1(A )) U Int (A) U aAA. The right side of (2.47) is the disjoint
union of two open sets. These two sets are nonempty because of (2.45),
(2.46), and the relations (which hold in every topological space)
(S\C1(A))U aA D S\A, Int(A)U aA DA.

16 Weak Convergence and Uniformity Classes
However this would imply that B (x : e) is not connected. We have reached
a contradiction. Hence xE(8A)`, and (2.40) is proved. The relation (2.41)
is therefore equivalent to (2.39). Q.E.D.
COROLLARY 2.7. Let S be a separable metric space. A class of
bounded functions is a Q-uniformity class for every probability measure Q on
S if and only if (i) sup {wt(S) : f E } < co, and (ii) is equicontinuous at
every point of S; that is,
lim sup w1(x : () = 0 for all x E S. (2.48)
JE J
Proof We assume, without loss of generality, that the functions in 15 are
real-valued. Suppose that (i) and (ii) hold. Let Q be a probability measure
on S. Whatever the positive numbers S and a are,
sup Q{{x:,1(x:e)>6))<Q({x: sup wf(x:e)>S}).t (2.49)
JE9J fE^
For every positive 8 the right side goes to zero as €,0. Therefore, by
Theorem 2.4, is a Q-uniformity class. Necessity of (i) also follows
immediately from Theorem 2.4. To prove the necessity of (ii), assume that
there exist a positive number S and a point xo in S such that
sup c j(xo:e)>S foralle>0.
fE9
This implies the existence of a sequence {x„) of points in S converging to
xo and a sequence of functions (f„) c F such that
IL(xn)?f.('XO)I>2
(n=1,2,...).
Let Q = Sxo and Q„ = Sx. (n =1, 2, ... ). Clearly,
(Q}
converges weakly to Q,
but
f
f.dQ,^?f.dQl =If^(x,^)?f,,(xo)l>2 (n=1,2,...).
Hence is not a Q-uniformity class. Q.E.D.
We have previously remarked that the weak topology on the set Vii' of all
probability measures on a separable metric space is metrizable, and the
tThe set (x: sup(wj(x : c) : f E f) >6) = u ((x : wj(x : c) > 8):f E i} is open (see Section 11)
and, therefore, measurable.

Uniformity Classes 17
Prokhorov distance dp metrizes it. Another interesting metrization is pro-
vided by the next corollary. For every pair of positive numbers c. d, define
a class L(c, d) of Lipschitzian functions on S by
L(c,d) = { f : wf(S) < c, I f (x) ? f (y)l < dp(x,y) for all x,y ES). (2.50)
Now define the bounded Lipschitzian distance dBL by
dBL (Q1, Q2) = sup
I
fdQ, ? f fdQ2I (Q1.Q2E''P). (2.51)
fEL(1,1)
COROLLARY 2.8. If S is a separable metric space, then dBL metrizes the
weak topology on
Proof. By Corollary 2.7, L(l, I) is a Q-uniformity class for every probabil-
ity measure Q on S. Hence if (Qn) is a sequence *of probability measures
converging weakly to Q, then
lira
dBL (Qn,Q)=0.
(2.52)
n
Conversely, suppose (2.52) holds. We shall show that (Qn) converges
weakly to Q. It follows from (2.52) that
lim l f fdQn? f fdQl =0 for every bounded Lipschitzian function f.
(2.53)
For, if f (x) ? f (y)^ < dp(x,y) for all x,y E S,
f fdQ,,? f fdQ=c( f f'dQn
? f f'dQ),
where c = max {wf(S),d) and f' = f/c E L(1, 1). Let F be any nonempty
closed subset of S. We now prove
lim Qn(F)<Q(F). (2.54)
n
For E >0 define the real-valued function fE on S by
ff(x)=^(e -'p(x,F)) (x ES), (2.55)

18 Weak Convergence and Uniformity Classes
where 4, is defined on [0, oo) by
(t) __1
. 1?t if 0 < t < 1,
(2.56)
0 if 1>1.
Note that f is, for every positive e, a bounded Lipschitzian function
satisfying
wf(S)<1, If,(X) ?f^(Y)I<)EP(x+F)? -p(Y,F)I
E
P(x,Y) (x
,Y E S ),
so that, by (2.53),
linm f f^ dQ„ = f f, dQ (e>0). (2.57)
Since IF f, for every positive e,

linm Q,, (F) < lim f f dQ„ = f f dQ
(e>0).
(2.58)
Also, lim f^(x)= IF(x) for all x in S. Hence
lim JJdQ=Q(F). (2.59)
cj0
By Theorem 1.1 { Q) converges weakly to Q. Finally, it is easy to check
that dBL is a distance function on 'P. Q.E.D.
Remark. The distance daL may be defined on the class GR, of all finite signed measures on S
by letting Qt. Q2 be finite signed measures in (2.51). The function ??.d,L(s,0) is a norm on
the vector space 9R.. The topology induced by this norm is, in general, weaker than the one
induced by the variation norm (1.5). It should also be pointed out that the proof of Corollary
2.8 presupposes metrizability of the weak topology on 9 and merely provides a suitable
metric as an alternative to the Prokhorov distance do defined by (1.16)]. This justifies the use
of sequences (rather than nets) in the proof.
One can construct many interesting examples of uniformity classes
beyond those provided by Corollaries 2.7 and 2.8. We give one example
now. The rest of this section will be devoted to another example (Theorem
2.11) of considerable interest from our point of view.

Uniformity Classes 19
Example Let S=R2. Let 6! (1) be the class of all Borel-measurable
subsets of R2 each having a boundary contained in some rectifiable curvet
of length not exceeding a given positive number 1. We now show that 6D (I)
is a Q-uniformity class for every probability measure Q that is absolutely
continuous with respect to the Lebesgue measure A2 on R 2. Let A E 6 (l)
and let 3A C J, where J is a rectifiable curve of length 1. There exist k
points zo,zt,.••,zk-, on J such that (i) zo and zk_t are the end points of J
(may coincide), (ii) k<I/c+2, and (iii) Iiz
1?z1_,II<E for i=1,2,...,k-1
(11.11 denotes euclidean norm). The k open balls having z;'s as centers and
radii 2E cover J`. Hence
A2((aA)`) <1A2(J (L +2)1T(2E)
2
=41rk+87rE2 [A E (l )]. (2.60)
Let Q be a probability measure that is absolutely continuous with respect
to A2. Then (2.60) implies (in view of Theorem A.3.1)

lim (sup{Q((aA):AE^'i)(1)})=0. (2.61)
By Corollary 2.6, -D (1) is a Q-uniformity class.
We need some preparation before proving the next theorem. Let S be a
metric space. Define the Hausdorff distance A between two closed bounded
subsets A, B of S by
A(A,B)=inf{ E:c>0,AcB`, BCA`}. (2.62)
The class Q of all closed bounded subsets of S is a metric space with
metric A.
LEMMA 2.9. Let ')1t be a compact subset of (? . Then for every probabil-
ity measure Q on S one has
lim sup{Q(M`):ME )=sup{Q(M):MElt). (2.63)
cjo
Proof Let rq be a given positive number. For every M E )lt. there exists a
positive number E,t, such that
Q(M`")<Q(M)+"1
,
to rectifiable curve in R2 is a subset of R2 of the form {z(t):0G t < 1), where t- z(t)
=(x(t),y(t)) is a continuous function of bounded variation on [0,11 into R2; z(0), z(1) are
called the endpoints of the curve.

20 Weak Convergence and Uniformity Classes
since M`J.M as 40. By compactness of Olt, there exists a finite collection
of sets {M1, ..., M,) in OL such that every set in OJlt, is within a A-distance
from M for some i, 1 <i < r. Let
EM;
Eo=-min { :1<i<r }.
Then
sup(Q(M`"):MEOL) <sup{Q(M,`°Mi 2):1<i<r}
<sup(Q(M iAli ): 1 <i <r)
<sup {Q(Mi)+rl: 1 < i < r}
<sup (Q(M): M E Olt,}+rl.
Q.E.D.
A subset C of Rk is convex if a x, + (1 ? a) x2 E C for all x,, x2 E C
and all a E [0,1]. A hyperplane H in R' is a set of the form
H={x:<u,x>=c}, (2.64)
where c is a real number and u is a unit vector of Rk; that is, Ilul) = 1, and
<,> denotes euclidean inner product
k
<u,X>= u,X,, (2.65)
i=!
k 1/2
(lull u? [u=(u1,...,uk), x=(x1,...,xk)ERk].
i^l
A closed half space E is a set of the form
E=(x:xER k,<u,x)<c) {cER', uER k, IIull=l]. (2.66)
A hyperplane H= (y : <u,y> = c) is said to be a supporting hyperplane for a
set A at x E A if
<u,x>=c, AC{z:<u,z><c}, (2.67)
that is, if the hyperplane H passes through the point x of A and has A on
one side. Note that the sign < in (2.66) and (2.67) may be replaced by >

Uniformity Classes 21
(by merely changing u to ? u). It is a well-known fact that if A is a
compact convex set, then there exists a supporting hyperplane for A at
each x E aA .t
LEMMA 2.10. Let A, B be two closed, bounded, convex subsets of Rk.
Then
A(A,B)=A(aA,aB). (2.68)
Proof. Suppose A(8A,aB)=E. Let E' be any number larger than E. Let
x E A. There exist xr, x2 E aA and a E [0,11 such that x = ax I +(1 ? a)x2,
since the intersection of A with any line through x is a closed line segment
whose end points (x,,x2, for example) lie on 8A. Let y,, y2 E aB be such
that xi ?y; < E' for i= 1, 2. Then letting y = ay, +(l ? a)y2 yields y E B and
llx?yl! <?Ilx1
?yell+(I
?
a)IIx2?y2fl
<E'.
Thus A c B` . Similarly B CA '. Since this is true for every E > E,
0(A, B) < A(aA, aB ). (2.69)
To prove the opposite inequality, suppose that E > t1(A, B) and let be any
positive number. Let x E aAA. Let (z : (1, z> = c) be a supporting hyperplane
for A at x. Then the half space H = { z : (l, z> c c + E) contains A. If z E A
and 11 z' ? z l l< E, then
(l,z'>=(l,z>+(l,z'?z> <l,z>+IIz'?z!I c+ e.
Hence H J A D B. The ball B (x : E + il) intersects R k H. This is because
the point x+(E+71/2)1 of this ball satisfies
(l,X+(E+ 2 )l>=(l,x>+(E+ 2)=C+E+ 2 ,
and therefore lies in the complement of H. It follows that B (x : E + rl)
intersects R k B. But, since A C B ` and x E A, B (x : E + rl) certainly inter-
sects B. It follows that B (x : E + rl) intersects aB, so that (aB)"'' D 3A.
Similarly (8A)`+' j B. Therefore
A(aA,aB)<E+rl
tEggleston II), p. 20.

22 Weak Convergence and Uniformity Classes
for every c > 0(A, B) and every positive rt. Hence
0(aA,aB)<0(A,B). (2.70)
The inequalities (2.69) and (2.70) together yield (2.68). Q.E.D.
THEOREM 1.11. Let C denote the class of all Bore/-measurable convex
subsets of Rk. Let Q be a probability measure on Rk. The class e is a
Q-uniformity class if and only if it is a Q-continuity class, that is, if and only
if
Q(ac)=o forallCEe. (2.71)
Proof If e is a Q-uniformity class, then, by relation (2.41) in Corollary
2.6, e is a Q-continuity class. Suppose, conversely, that (2.71) holds. Since
the characterization (2.41) of Q-uniformity is in terms of boundaries, and
since aC= a (Cl(C)) for every convex set [note that C1(C)= Int(C)u aC
and that aC has empty interior for convex C], it follows that C is a
Q-uniformity class if and only if e = (C 1(C) : CE C) is. Given ri >0, let r
be so chosen that
Q({x:IIxII>r))<2. (2.72)
Write
C', = { C: C E e, CCC1(B (0: r))).
By a well-known theorem of Blaschket 3, is compact in the Hausdorff
metric A defined by (2.62). Lemma 2.10 shows that the compactness of ,
is equivalent to the compactness of { aC : C E 13, }. Lemma 2.9 and the
hypothesis (2.71) now yield
lim sup{ Q((8C)`):CEC',}=0. (2.73)
By Corollary 2.6, C, is a Q-uniformity class. Let (Qn) be a sequence of
probability measures weakly converging to Q. Then
lim (sup{IQn(C)?Q(C)I:CEC^))
n
<11im (sup{IQ.(C)?Q(C)^:CE,})
+ lim Qn({x:llxll>r))+Q({x:llxll>r))
n
=2Q({x:fIxll >r})<ii. (2.74)
tSee Eggleston [1), pp. 34-67.

Integrals Over Convex Shells 23
Note that the last equality follows from the fact that (x : Ilxil > r) is a
Q-continuity set, since its complement is [although, given any probability
measure Q and a positive rl, one can always find r such that (x: jlxjl > r) is
a Q-continuity set and (2.72) holds]. Since rl is an arbitrary positive
number, it follows from (2.74) that a is a Q-uniformity class. Con-
sequently, (2 is a Q-uniformity class. Q.E.D.
Remark. It follows from the above theorem that if Q is absolutely continuous with respect
to Lebesgue measure on R", then e is a Q-uniformity class. In particular, a is a 4)-
uniformity class, 4) being the standard normal distribution in Rk. We shall obtain a
refinement of this last statement in the next section.
It is easy to see from the proof of Theorem 2.11 that the class C in its statement may be
replaced by any class Ef of Borel-measurable convex sets with the property that
R,-(CI(A)nCl(B(0:r)):AE($}
is compact in the Hausdorff metric for all positive r. In particular, by letting
d ? {(?oo,x1]X(?oo,x2]X ... X(?oo,xk]:x?(xi,...,Xk)ERk },
we get Polya's result: Let P be a probability measure on Rk whose distribution function F,
defined by
F(x)-P((-oo,x,]x... x(-oo,xk]) [x-(x......xk)ERk ], (2.75)
is continuous on Rk. If a sequence of probability measures {P„} with distribution functions {FA)
converges weakly to P, then
sup(IF„(x)-F(x)I:xERk }-+0 (2.76)
as n-boo. The left side of (2.76) is sometimes called the Kolmogorov distance between Po and P.
The converse of this result is also true: if (2.76) holds, then (P„) converges weakly to P. In
fact, if (F„} converges to Fat all points of continuity of F, then (P„) converges weakly to P.t
3. INEQUALITIES FOR INTEGRALS OVER CONVEX SHELLS
It is not difficult to check that if C is convex then so are Int(C),C1(C).
The convex hull c(B) of a subset B of Rk is the intersection of all convex
sets containing B. Clearly c(B) is convex. If C is a closed and bounded
convex subset of Rk, then it is the convex hull of its boundary; that is,
c(8C)=C.
Clearly c(8C) C C. On the other hand, if x E C, x li= 8C, then every line
through x intersects aC at two points and x is a convex combination of
these two points. Thus c(aC) = C. If C is convex and e >0, then C` is
tSee Billingsley [11, pp. 17-18.

24 Weak Convergence and Uniformity Classes
convex and open, and C -' is convex and closed. The main theorem of this
section is the following:
THEOREM 3.1. Let g be a nonnegative differentiable function on [0, oo)
such that
(i)b= f Ig'(t)Itk-'dt<cc,
0
(ii)lim g(t)=0.
r--►oo
Then for every convex subset C of Rk and every pair of positive numbers e, p,
g(JI xI1)dx < bak(E+p)
(3.1)
C'\C-v
where
k7l
k/2 2^7k/2
ak
_
f((k+2)/2) I'(k/2) '
(3.2)
is the surface area of the unit sphere in Rk.
COROLLARY 3.2. Let s > 0, k > 1, and
f(x)=(21T)-k/2IIxIIsexp j ?
11x112
1
(xERk )
Then for all convex subsets C of Rk and every pair of positive numbers e, p,
f
c: '/2(2s
r((k+s-1 )/2)
f(x)dx<2 -+k-1)
r(k/2)
((+p). (3.3)
C.\C -,
Proof. Here one takes (in Theorem 3.1)
g(t)=(217)-k/2 t'exp(-12/2) 1E[0,00).
Then
g,(t)=(2r)-k/2(st$-I? is+I
)exp (? 2 1,
( )?k/2
fo
00
( k+s-2 k+s) p r ? 2 l
dib < 2^r st + t ex j 2 }
l J

Integrals Over Convex Shells 25
=(2.r)-k/2(s,2(k+s-3)/zr(
k+s? 1
)+21c+3_/2r(
k+s+ 1
2 2
k/2 (k+s-3)/2
=(2^r)- 2 (2s+k-1) r(
k+s - 1
2
=2(s-3)/2(2s+k? I) -k/2 r(
k+s-1)
(3.4)
which gives (3.3) on substitution in (3.1). Q.E.D.
The rest of this section is devoted to the development of the material
needed for the proof of Theorem 3.1.
For k=1, C`\C -° is contained in the union of two disjoint intervals
each of length e+ p. Hence
f
g(IxI)dx 2(e +)( sup g(x)) <2(e+p) f 'QI g'(t)I dt=2(e+p)b
C`\C-^ x>0 0
=ba1(e+p).

(3.5)
For k >. I a more intricate argument is needed. For the rest of the section
we assume k> 1. A polyhedron is a closed, bounded, convex set with
nonempty interior that is the intersection of a finite number of closed half
spaces. If P is a polyhedron, a face of P is a set of the form H n aP, where
H is a hyperplane such that H n aP has nonempty interior in H.
LEMMA 3.3. Let a polyhedron P be given by
P=(x:<u^,x)<d^, 1 < j<m}, (3.6)
where ui's are distinct unit vectors. Let
Li= (x:xEP, <u^,x>=d) (I < j<m). (3.7)
Then a P = U L^. If F is a face of P, then F = L^ for some j. Moreover,
P= (x : (u^,x> < d for all j for which LL is a face of P }. (3.8)
Proof. The first assertion is obvious. If F= H n 8P= u(H n L^) is a
face of P, then H = (x : <u^, x) = d) = H^, say, for exactly one j, since
the intersection of H with any hyperplane distinct from H has empty
interior in H. This proves the second assertion. For the third, note that

26 Weak Convergence and Uniformity Classes
Lj = Hj n aP. It is clear that the interior of L1 in H^ is
(x:<u,,x><d, for r#j, <u,,x>=d ),
so that if Lj is not a face of P, then Lj C U L,. This implies 8P C 8Q,
r#j
where Q is defined by
Q= { x : <uj,x> <d for allj for which L1 is a face of P).
Clearly P c Q. It is sufficient to show that aP c aQ implies P= Q. Let
xt E Int (P). Assume there exists x2 E aQ aP. Consider the line segment
[x1,x21 joining x, and x2. This line segment meets aP at x3, say. Clearly,
[x1,x2)CInt(Q) and x3 x2, so that x3EInt(Q)naP, which contradicts
the fact aP c Q. Q.E.D.
LEMMA 3.4. A polyhedron P has a finite number of faces. if F1,.. .,Fm
are the faces of P, then a P = U F. Moreover, there exist unique unit vectors
u^ and constants d3 such that F, = (x : x E P, <u^, x> = dd ), and P C (x : <u^, x>
< di). Also, P then has the representation
P=(x:<uj,x><dj, l <j<m}. (3.9)
Proof. This lemma follows easily from Lemma 3.3. Note that the repre-
sentation (3.9) is unique up to a permutation of unit vectors uj's and the
corresponding constants d's. Q.E.D.
Remark. A polyhedron is the convex hull of a finite number of points. In fact, since each
face is a polyhedron in a lower dimensional affine space, this follows by induction on k.
Conversely, it is known that the convex hull of a finite set of points can be expressed as the
intersection of a finite number of closed half spaces.t
There are two main steps in the proof of Theorem 3.1. The first one
(Lemma 3.9) is to express the surface integral as a derivative of volume in
volume integrals. The second step (Lemma 3.10) is to get a uniform bound
for the surface integral of a fixed function over the boundary of a
polyhedron. The ideas involved here belong naturally to the domain of
surface area and surface integrals. In the following paragraphs we develop
the material needed for the proofs. The development here is self-contained
except for the use of Cauchy's formula, which is stated but not proved.
We begin with some notation. Let Ak denote the Lebesgue measure on
R k normalized by the euclidean distance on R k; that is, if y1, ... ,yk is an
tSee Eggleston 111, pp. 29-30.

Integrals Over Convex Shells 27
orthonormal basis and A is a cube with respect to them or A = { I ty; : a;
< t; < b; for all i}, then X.(A)=(bt ? at)• • • (bk ? ak). We also refer to Ak as
the k-dimensional Lebesgue measure on R'. On any hyperplane of R'
there is a (k ? 1)-dimensional Lebesgue measure normalized the same
way. We denote this measure as Ak _ 1 and call it the (k? 1)-dimensional
Lebesgue measure. For example, if H is a hyperplane, it can be written in
the form H=xo+Ry1+---+Ryk_1, where
x0,y1,...,yk-1EH and
y,,•..,yk_I are (k?I) orthonormal vectors in R'. If f is a function with
compact support in R k, then
f
HfdXk-1= f f(xo+t1Y1+
...
+tk-1Yk-1)dt,...dtk_1. (3.10)
Next we denote ak_l as the surface area measure on the unit sphere
Sk_ . One can write an explicit formula for
0k- 1 by using Eulerian angles
to parametrize points of Sk _ 1.
It then follows that ok_I(Sk_1)=ak, where a, is given by (3.2).
Let P be a polyhedron with faces F, (1 < i < m). Then the surface area of
P is defined as
Ak_1(8P)= Ak-1(F;). (3.11)
;=t
For a bounded Borel-measurable function f on R k we define the surface
integral off on a P by
faP
fdXk-1= ^, f fd^k-1
(3.12)
and if A is a Borel subset of aP, then the surface area of A is defined by
Ak-1(A)=

Ak-1(AnF;).
(3.13)
Remark. Let P be given as P={x:<u^,x><4., I <j< m}. Then
f
fdak_1= f fdAk_1. (3.14)
aP I<j<m L,
where LL-(x:xEP,<u 1,x>=d1). This follows from the fact that A1(L)0 if L^ is not a
face of P.

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Documents with Different Content

Betaught, [bi-tahte] == taught. Rel. S. v. 124
Bete, v. a. lit. == ‘make better;’ hence, ‘heal,’ ‘save.’ Marg. 68
—— == ‘recompense,’ ‘make amends for.’ RG. 369. AS. bétan
Bete, part. == beaten. Vid. Beat
Beten, [y-beten] == overlaid, covered, as with silk, gold, &c. Alys.
1034, 1518
Beth, Beoth, &c. See Be
Bethink, v. a. == ‘to bethink oneself’ of a thing. RG. 368, 458
Betide, v. n. == happen. RG. 418, 14
Betime, adv. K. Horn, 995
Betoken, v. a. RG. 152
Betokening, sb. RG. 560
Betray, v. a. RG. 135
Better, adj. RG. 367, 422
—— v. n. == get the better. Ps xii. 5
Betterness, sb. Ps. li. 5
Between, prep. RG. 371, 513
Betwixt, prep. [bi-tuxen]. O. and N. 1745
Beverage, sb. == drink. RG. 26
—— == reward, consequence. RG. 299
Bewail, v. a. Alys. 4395
Beware, v. n. RG. 547

Beweep, v. a. O. and N. 972
Bewind, v. a. == entwine. part. ‘bewound.’ Christ on the Cross, 3
Bewray, v. a. == betray [by-wrye]. Alys. 4377. pret. ‘bi-wro.’ O. and
N. 673. AS. wrégan.
Beyen, == are? Wright’s L. P. p. 32
Beyond, prep. RG. 368, 420
Beyre, == of both, gen. pl. RG. 388, 398
Bezant, sb. == a piece of money. RG. 409. From Byzantium, or
Constantinople, where they were originally used
Bible, sb. Rel. Ant. ii. p. 174
Bicast, v. a. == cast over, cover. 92 β
Bicatch, v. a. == deceive, ensnare. Alys. 258. K. Horn, 318
Bicharred, part. == deceived. Rel. Ant. ii. p. 211; M. Ode, 160. AS.
becýrran
Bicherme, v. a. == chirp about or around. O. and N. 279. AS. cyrm
Bick, v. n. == fight. Alys. 2337
Bicker, v. n. == quarrel. RG. 540. Fr. becquer. W. bicra == to fight
Bicker, sb. == a quarrel, contention, battle. RG. 538, 543
Biclipe, Biclupe, v. a. == accuse. 365 B.
—— == appeal. RG. 473
Biclose, v. a. == enclose. RG. 558, 218
Bid, v. a. == ask. RG. 77. 3 pl. pret. ‘badden.’ Alys. 5823. See ‘bede’
—— == command. RG. 29. pret. ‘bad.’ 683 B. part. ‘y-bede.’ RG.
383. AS. biddan

Bid, v. a. == offer, pret. ‘bode.’ RG. 379. ‘beod’? O. and N. 1435. AS.
beódan
Bid, sb. == asking, demand. Pol. S. 149
Bidding, sb. == demand, request. Pol. S. 150
Bide, v. n. == remain. Pol. S. 204
Bidene, adv. == presently. Ps. l. 4; ciii. 30
Bidelve, v. a. == bury. Rel. Ant. i. 116
Bidone, part. == ‘bidun in grave.’ Body and Soul, 97
Bier, sb. 128 B.
Bieren, sb. == a man. Ps. cxxvi. 5; cxxxix. 2. AS. beorn
Biflette, v. n. == flow past. K. Horn, 1457
Bifluen, v. a. == flee from. M. Ode, 77
Big, v. a. == build. Ps. xxvii. 5. AS. byggen. ON. byggja
Bigabbed, part. == deceived. Lit. ‘talked over.’ RG. 458. AS. gabban
Bigate, sb. == booty. Alys. 2138
Biggand, sb. == a builder. Ps. cxvii. 22
Biglide, v. n. Wright’s L. P. p. 87
Bigrede, v. a. == lament. Alys. 5175. AS. grædan.
—— == call to. O. and N. 279
Bihaite, v. a. == behold? O. and N. 1320. AS. behawian. Or,
possibly, == observe, regard. AS. hedan. Germ, behüten. See Gloss.
Rem. on Laȝ. iii. 457
Bihalves, adv. == aside. St Kath. 13

Bihede, v. a. == regard. O. and N. 635
Bihemmen, v. a. == cover, cloak. O. and N. 672
Bihepe, part. == heaped up. O. and N. 360
Bihete, v. a. == promise, pret. ‘bihet.’ RG. 381. ‘byheyghte.’ Alys.
3926
Bihoting, sb. == promise. Alys. 4000
Bike, sb. == cassia. Ps. xliv. 9. Literally ‘pitch.’ ON. bik
Bilace, part. == beset. Alys. 3357
Bilaue. See Bileve
Bilaucte. See Bilou
Bilede, v. a. == lead about. Pol. S. 155. O. and N. 68
Bilegge, v. a. == assert, allege. O. and N. 672
Bileve, v. a. == leave. RG. 421
—— v. n. == remain. RG. 372, 374. [bilaue]. Alys. 3541
Biliked, part. == rendered likely or probable. O. and N. 840
Bilime, v. a. == to mutilate. RG. 471, 560
Bilimp, v. n. == happen. M. Ode, st. 59 AS. belimpan
Bill, sb. (of a bird). O. and N. 79
—— == hatchet. Pol. S. 151
Bilou, pret. == laughed at. RG. 328. [bylaucte]. K. Horn, 681. [by
lowe], RG. 299. [by lowȝ]. RG. 64
Bimong, prep. == among. Wright’s L. P. p. 35
Bind, v. a. Wright’s L. P. p. 45. part. ‘ibounde.’ RG. 487

Binder, sb. HD. 2050
Binding, sb. == chain. Ps. cxxiv. 5
Bink, sb. See Bench
Bipahte, pret. == deceived. Rel. S. v. 128. AS. be-pæcan
Birade, v. a. == counsel. Alys. 3732
Birch, sb. == the tree. Alys. 5242
Bird, sb. RG. 177
Birde, sb. == lady. HD. 2760. A metathesis of ‘bride’
Birst, == bruised. Body and Soul, 86. AS. berstan.
Birth, sb. == nation. Ps. lxxviii. 10
Birthman, sb. == man of good birth. HD. 2101
Birthtime, sb. [burtyme]. RG. 9, 443
Birue, v. a. == rue, repent. Fragm. Sci. 325
Bis, sb. == purple. Wright’s L. P. p. 26. Fr. bis. Lat. byssus
Bisay, v. a. == recommend, say. RG. 422
Bisayen, == treated. See Besee.
Bischriche, v. a. == shriek at. O. and N. 67
Biscunien, v. a. == shun. M. Ode, 77
Bise, sb. == north wind. HD. 724. OHG. bísa
Bisend, v. a. == send after. RG. 491
Bishop, sb. RG. 376
Bishopric. RG. 414, 417

Bismere, sb. == blasphemy. Body and Soul, 110. [busemere]. RG.
12, 379. AS. bismér
Bisne, adj. == blind. O. and N. 78. AS. bisen
Bisoht, == sought out, got ready for. Pol. S. 220
Bisokne, sb. == beseeching. RG. 495
Bispel, sb. == proverb. O. and N. 127. AS. bispel
Bistad, sb. == a dwelling. Wright’s L. P. p. 38
Bistand, v. a. == stand by a person; hence, to press or urge them.
O. and N. 1436
Bistolen, part. == stolen, crept onwards. M. Ode, 9
Bisyhed, == the state of being busy. Alys. 3
Bit, sb. == a morsel. RG. 207
Bit, sb. == bottle. Ps. lxxvii. 13. [bite]. Body and Soul, 34. AS. bitte
Bitch, sb. Alys. 5394
Bite, v. a. Alys. 5435
Bite, sb. Alys. 5436
Bite, v. a. == drink. HD. 1731. Cf. bohem. ‘piti,’ potus; ‘pitka,’
potatio, &c. Gr. πίνω
Bitell, v. a. == excuse. O. and N. 263
Bitiȝt, == arrayed. O. and N. 1011. AS. biþæht. See Gloss. to Laȝ.
s. v.
Bitoȝe, == employed. O. and N. 702. AS. biteon. See Gloss. to Laȝ.
s. v.
Bitter, adj. Wright’s L. P. p. 87

Biturn, v. a. == turn. RG. 210
Bituxen. See Betwixt.
Biwene, v. a. == discover, recognize. O. and N. 1507
Biwente, vb.—‘hire bi-wente.’ == turned her about. K. Horn, 329.
In pass. ‘þai bewent’ == let them be turned back. Ps. vi. 11. AS.
wendan
Biwere, v. a. == protect. O. and N. 1124. AS. bewerian
Biweved, == covered. RG. 338.
—— == woven? Alys. 1085
Biwin, v. a. == win. RG. 75, 420
Biwit, adv. == out of one’s wits. RG. 528
Biwite, v. a. == defend. Rel. S. v. 252. AS. bewitan
—— == know? Alys. 5203
Biwrye, v. a. == cover. Alys. 6453. AS. wreon.
Black, adj. RG. 433, 522
Blacken, v. n. == become angry. HD. 2165
Blame, v. a. RG. 163
—— sb. RG 272, 432
Blandishing, sb. == blandishments. St Kath. 164
Blanis ? Alys. 6292
Blanket, sb. 1167 B. Fr. blanchet
Blast, v. n. == blow, puff. Alys. 5349
Blast, sb. Fragm. Sci. 190. Ps. cxlviii. 8

Blaze, sb. 1254 HD. AS. blǽse, blýsan
Blear, v. n. == become bleareyed. Rel. Ant. ii. p. 211
Bleat, v. n. Ps. lxiv. 14
Bled, blete, sb. == foliage. O. and N. 1040, 57. AS. blæd
Bleed, v. n. RG. 560
Bleike, adj. == pale. HD. 470. AS. blác. ON. bleikr
Blench, sb. == a trick? O. and N. 378. ON. blekkja
Blench, v. n. == avoid (a thing). O. and N. 170
—— == flinch from [blinche]. 2184 B.
—— == deceive. Ritson’s AS. viii. 23
—— == give way? (of a ship) K. Horn, 1461. Another form of ‘flinch.’
AS. blinnan
Bleo, sb. == hue, complexion. O. and N. 152. Wright’s L. P. p. 35.
AS. bleo
Bless, v. a. RG. 406
Blessing, sb. RG. 421
Blete, adj. == bleak? O. and N. 616
Blete, sb. See Bled
Blike, v. n. == shine. Wright’s L. P. p. 52 AS. blícan
Blinch. See Blench
Blind, adj. RG. 376, 407
—— v. n. == become blind. Rel. Ant, ii. p. 211

Blink, sb. ‘to make blinks,’ == deride a person. HD. 307. See
Blench, sb.
Blin, v. n. == cease. RG. 566. pret. ‘blenyte.’ RG. 338. AS. blinnan
Bliss, sb. RG. 469
Blissful, adj. Wright’s L. P. p. 52
Blissfully, adv. Ps. xcvi. 1
Blithe, adj. RG. 15
Blithely, adv. 89 β
Blitheful. Ps. cxi. 5
Blive, adv. == quickly. RG. 544. See Belive
Blode, adj. == pale, dried up. Rel. Ant ii. p. 210. Germ. blöde
Blood, sb. RG. 388, 416
Bloody, adj. RG. 304, 311
Bloom, sb. HD 63
Bloom, v. n. Ps. xxvii. 7
Blote, adj. == dried. Rel. Ant. ii. p 176
Bloute, v. n. == swell out? HD. 1910. ON. blautr. Eng. bloat
Blow, v. a. == as ‘blow the fire.’ HD. 385. Alys. 5030
—— v. n. pret. ‘blew.’ 524 β
Blow, vb. n. part. ‘blowe,’ == blown, in blossom. O. and N. 1634
Blowing, sb. 467 β
Blue, adj. [blo]. Wright’s L. P. p. 86

Bo, == be. O. and N. 166, et passim. See Be
Bo, == both. q. v.
Boar, sb. RG. 133
Board, sb. == table. 122 β.; plank. Alys. 6415
Boast, sb. RG. 258. pomp. St Swithin, 43
Boast, v. n. Alys. 2597
Boasty, adj. == boastful. Fragm. Sci. 283
Bobance, sb. == boasting. Pol. S. 189 Fr. bobance
Bochevampe, (sic in MS.). == botched vamps or fronts of shoes.
Rel. Ant. ii. p. 176
Bode, sb. == commandment. Ps. cxviii. 134, 128, et passim
Bode, v. a. == foretell. O. and N. 530
Boded ? Pol. S. 152
Boding, sb. RG. 416, 428
Bodeword, sb. == message. Ps. ii. 6
Body, sb. RG. 395, 547
Boffing, == swelling or puffing. RG. 414. Fr. buffer, to puff the
cheeks
Boistous, adj. == coarse, rude. Alys. 5660. [boustes] Fragm. Sci.
273
Bold, sb. == a building. RG. 44. AS. bold
Bold, v. a. == embolden. Alys. 2468. [bald]. Ritson’s AS. viii. 128
—— adj. RG. 383. ‘bolder.’ RG. 465

Boldhede, == boldness. O. and N. 514
Boldly, adv. RG. 500, 19
Boleax, sb. == large axe. Rel. Ant. ii. p. 176. ON. bolöxi
Bolken, v. n. == belch. Ps. cxliii. 13
Bollen, == swollen. Body and Soul, 31. ‘ibolȝe.’ O. and N. 145
Bolster, sb. Rel. S. v. 90
Bolt, sb. ‘ȝoure bolt is sone ischote.’ St Kath. 54
Bonde, sb. == bondman. Pol. S. 150
Bondman. RG. 370. HD. 32
Bone, sb. == os. RG. 446
Bone, sb. == prayer. RG. 14. AS. bén. SS. bone
Boned, [y-boned] == having bones. RG. 414
Bonére, adj. == debonair, graceful. Alys. 6732
Bonny, adj. Alys. 3903
Book, sb. RG. 374, 420
Boot, sb. == use, avail. Body and Soul, 92
—— == remedy, means (bote). RG. 277, 408. Pilate, 139
Booth, sb. Alys. 3457
Booze, sb. [bous] == drink. Wright’s L. P. p. 111. Dutch, buysen
Booze, v. n. == drink. Rel. Ant. ii. p. 175
Bord, sb. == border. Alys. 1270
Borough, sb. [boru]. RG. 72

Borow, v. a. == defend. Wright’s L. P. pp. 24, 25. part. ‘iborȝe,’ O.
and N. 881
Borow, sb. == surety. RG. 472, 497
Borrow, v. a. RG. 393
Borstax, sb. == pick-axe. Pol. S. 151
Bosk, sb. == wood. RG. 547. Fr. bos, bosche
Boss, sb. == an ornament of dress. Pol. S. 154. Fr. bosse
Bote, sb. See Boot
Botemay, sb. == bitumen. Alys. 4763
Botfork, sb. == a crooked stick. Wright’s L. P. p. 110
Both, adj. RG. 376, 445. ‘both two.’ Body and Soul, 120. [bo].
Wright’s L. P. p. 58
Both, == are. See Be
In O. and N. 630, 633, the meaning of ‘both’ is uncertain; perhaps a
mistake for ‘doth’
Botheler, sb. == peasant, shepherd. Body and Soul, 144; from
‘booth’?
Boting, sb. == recompense. Alys. 5711
Bough, sb. [bowe]. RG. 283. [boye], O. and N. 15
Bouk, sb. == body. Alys. 3946. [buc], O. and N. 1130. AS. búce.
Germ. bauch
Bouked, adj. == protuberant. Alys. 6265
Boulder, sb. == a large stone. HD. 1790

Boun, adj. == ready. Wright’s L. P. p. 100. Ritson’s AS. viii. 149. ON.
búinn.
Bound, sb. == boundary. Alys. 5593
Bouning, == making ready. Wright’s L. P. p. 25
Bout, sb. == apparently some female ornament for the face. Pol. S.
154
Bow, sb. RG. 377, 541
Bow, v. a. == bend. pret. ‘buyede.’ RG. 475. ‘beh.’ Wright’s L. P. p.
54. ‘bed,’ 2127 B.
—— v. n. == bow or bend. Wright’s L. P. p. 70. AS. búgan.
Bowels, sb. Pol. S. 213. Alys. 4668. For the etymology of this word,
see Phil. Soc. Trans. for 1856, p. 36
Bower, sb. HD. 2072. Wright’s L. P. p. 114. AS. búr.
Bowermaiden, sb. == Rel. Ant. ii. p. 175
Bowiar, sb. == bow-maker. RG. 541
Bowl, sb. K. Horn, 1155
Bowman, sb. RG. 378
Bowshot, sb. Alys. 3491
Box, sb. RG. 456
Boy, sb. Pol. S. 237
Boy, == man. HD. 1899
Brag, adj. == boastful, bold. Wright’s L. P. p. 24
Braid, vb. The following analysis of this difficult verb is taken from
Egilson’s Lex. Poet. Septent. s. v. bregða. All the senses here given

are found in the O. Norse, while the AS. ‘bredan’ apparently is only
used in those marked with an asterisk.
* I. act. to weave, part. ‘broiden.’ O. and N. 645
II. act. to move a thing from its place. Hence,
α. to draw out, as a sword. HD. 1825. part. ‘ybrad’ == drawn,
caught. Wright’s L. P. p. 39
β. to brandish, as a sword or spear. Alys. 7373
γ. to pull down. RG. 22. [breide], Alys. 5856
* δ. to seize, or perhaps tear. Rel. S. v. 200. [brede]
III. neut. to change, as—
α. to awake out of sleep. HD. 1282
β. of any violent motion of body, as to leap. Body and Soul, 46
Braid, sb. ==
1. a quick motion, from III. β.; hence, ‘at a breid’ == in an instant.
Body and Soul, 182. ON. bragð.
2. a violent struggle or wrench. RG. 22
Brain, sb. RG. 49, 446
Branch, sb. RG. 152
Brand, sb. == a burning mass. Body and Soul, 208
—— == torch. Alys. 5295. [brond]. AS. brand
—— == fire. Alys. 1856. [wilde bround]
Brased, adj. == of brass. Ps. cvi. 16
Brass, sb. RG. 2, 251

Bray, sb. == noise. Alys. 2175
Breach, sb. [bruch]. Wright’s L. P. p. 30
Bread, sb. RG. 238
Breadth, sb. [brede]. RG. 385
Break, v. a. 47 B. part. ‘i-broke’ 1005 B.
—— v. n. pret. ‘brake.’ 2154 B.
—— == to break out (of flesh). 2421 B.
Breaking, sb. == breach, gap. Ps. cv. 23
Breast, sb. RG. 419
Breath, sb. Fragm. Sci. 203
Breathe, v. n. Fragm. Sci. 202
Breche, == beech? q. v.
Breech, sb. == rump. RG. 322
—— == breeches. 260 B.
Breed, v. a. (of a bird). 2 s. pres. ‘breist.’ O. and N. 1631. RG. 177.
part. ‘ibred’ == brought up, educated. O. and N. 1722. Body and
Soul, 81
Breed, v. n. == spring forth. Wright’s L. P. p. 45
Breist, == breedest. See Breed
Breme, adj. == glorious, renowned. Wright’s L. P. pp. 52, 32. AS.
breme
—— == eager, lustful. O. and N. 202
Brenne, sb. == burning. HD. 1239

Breth, sb. == wrath. Ps. ii. 5; vi. 2. ON. brædi == anger
Breven, v. a. == write down. Pol. S. 156.
Brew, v. a. [browe]. RG. 26
Brewer, sb. Rel. S. vii. 35
Brewster, sb. Rel. Ant. ii. p. 176
Breȝe, sb. See Brow.
Breze, sb. == gadfly. Ps. civ. 34. AS. brimse
Briar, sb. RG. 331
Bridal, sb. Alys. 1071. K. Horn, 1064
Bride, sb. HD. 2131
Bridegroom, sb. [bridegome]. Ps. xviii. 6
Bride, sb. == bridle. Alys. 7626
Bridge, sb. RG. 399
Bridle, sb. RG. 396
Bright, adj. HD. 2131. Wright’s L. P. p. 33
Brim, sb. == brink. 476 β
Brimstone, sb. Body and Soul, 219
Bring, v. a. RG. 379. pret. ‘brought.’ RG. 309. part. ‘ybroȝt,’ ‘ibrouȝt.’
RG. 376, 491
Brinie, sb. == cuirass. HD. 1775. Fr. brugne, brugnie. The root is
‘brun’ from ‘brinnan,’ to burn or shine; Cf. OHG. brunna
Brink, sb. Alys. 3491. K. Horn, 147
Brise, v. a. == bruise. HD. 1835

Bristle, sb. Alys. 6621
Bristled, adj. == having bristles. Alys. 5722
Britheling, == worthless, a rascal. Rel. S. vi. 11. Cf. O. Eng.
‘brothell’
Brittene, == cut in pieces? HD. 2700. Cf. ‘brittned,’ in Gloss. to
Ormulum. AS. bryttian
Broach, sb. (an ornament). RG. 489. Alys. 6842
Broad, adj. RG. 1. [brede], O. and N. 963
—— v. a. == make broad, part. ‘ibroded.’ O. and N. 1310
Broerh, adj. == brittle? Wright’s L. P. p. 23
Brood, sb. RG. 70
Broodful, adj. Ps. cxliii. 13
Brook, sb. RG. 80
Broom, sb. (genista). Alys. 2492
Brost, sb. O. and N. 976, a mistake for ‘prost,’ i.e. ‘priest.’ The Jesus
Coll. MS. reads ‘preost’
Broth, sb. RG. 528
Brother, sb. RG. 371, 478
Brouke, v. a. == use, enjoy. HD. 311 AS. brúcan. Germ. brauchen
Brow, sb. Wright’s L. P. p. 28. [breȝe]. Ib. p. 34
Brown, adj. RG. 429
—— v. n. == become brown. Alys. 3293
Brun, sb. == a brown jar. K. Horn, 1134

Brune, sb. == a burning. O. and N. 1153
Brust, adj. == rough, brusque. Pol. S. 151
Brut, adj. == rough? RG. 536
—— == bright. Body and Soul, 57
Bruthen, adj. == fierce, fiercely boiling, ‘a bruthen led.’ Rel. S. v.
242. Connected with ‘breth,’ and AS. brédan, to warm
Bu, sb. == buffalo. Alys. 5957
Bu, vb. See Buy
Buck, sb. Ritson’s AS. iii. 8
—— == he-goat. Ps. xlix. 13
Buckle, sb. Wright’s L. P. p. 35
Buckler, sb. Alys. 1190
Budel, sb. == messenger. O. and N. 1167. Wright’s L. P. p. 22. AS.
bydel
Bugging, sb. == a building, or lodging. Pol. S. 151. AS. byggan.
ON. byggja
Bugle, sb. == buffalo. Alys. 5112
Buglehorn, sb. Alys. 5282
Build, v. a. RG. 439
Bulge, sb. == a lump, hump. Body and Soul, 185
Bulies, == bellows, q. v.
Bull, sb. (animal). RG. 116
Bull, (Pope’s bull). RG. 473, 494

Bullock, sb. Ritson’s AS. iii. 8
Bunting, sb. (the bird). Wright’s L. P. p. 40
Burde, sb. == beard. Alys. 1164
Burdon, sb. == a pilgrim’s staff. K. Horn, 1093. Fr. bourdon
Burel, sb. == sackcloth. Alys. 5475. Pol. S. 221. Fr. bure, burel. See
Roq.
Burgess, sb. RG. 540, 541
Burial, sb. See Buryel
Burn, v. a. pret. ‘barnde.’ RG. 380, 511. ‘brende.’ RG. 536. part.
pres. ‘berninde.’ RG. 534
Burn, sb. == rivulet. O. and N. 916. AS. byrnan, to burn. Cf. Lat.
torrens, from torreo
Burst, v. n. pret. ‘barst.’ RG. 437
Burst, sb. == injury. Wright’s L. P. p. 24. AS. byrst
Burthen, sb. HD. 807
Bury, v. a. RG. 123. part. ‘y-bured.’ RG. 382. AS. byrgan
Burying, sb. RG. 382. [beryng]. Alys. 4624
Buryels, sb. == a tomb, grave. RG. 204. AS. byrgels
Busemere, == blasphemy. See Bismere
Busily, adv. Ps. cxlii. 7
Busk, v. a. == array. Pol. S. 239
Busy, adj. Alys. 3906
But, adv. 43 B.

But, prep. == except [bote]. RG. 382. [butent]. Rel. S. ii. 25
—— == without [bute]. O. and N. 184. AS. bútan
But, sb. == a put, i.e. cast or throw, HD. 1040
But, part. == contended. HD. 1916. Fr. bouter
Butcher, sb. Pol. S. 192
Bute, prep. See But
Butler, sb. RG. 187, 438
Butte, sb. == a fish, probably a turbot. HD. 759. The Prompt. Parv.
translates it by ‘pecten;’ the Pictorial Vocab., published by Mr Wright,
p. 254, has ‘hic turbo’ == ‘a but.’ See N. and Q. 2d S. vi. 382. Sw.
butta
Butter, sb. HD. 643
Button, sb. Pol. S. 239
—— v. n. == break out. St Swithin, 151. Fr. boutonner. Cotgr.
Buxomness, sb. == obedience. RG. 234, 318. AS. buhsomnes,
from ‘bugan,’ to bow
Buy, v. a. [biggen]. Moral Ode, st. 33. [buggen]. O. and N. 1366.
pret. ‘bouȝte.’ RG. 379, 496. ‘bu,’ imper. RG. 390
—— == to exact atonement for. K. Horn, 912
—— == redeem. Ps. xxv. 11
Buyer, sb. == redeemer. Ps. xviii. 15
Buzzard, sb. Alys. 3049
By, prep. == beside (of place). 1213 B. ‘Nolde God that ich bi thé
sete’

—— == according to. 169 B. ‘bi his rede.’
—— == during (of time). 649 B. ‘bi myn ancestors daye.’ 2498 B. ‘bi
a Tuesdai’
—— == against. 871 B. ‘bi the Bischop of L. thulke word he sede.’
Cf. 1 Cor. iv. 4
—— == concerning, of. O. and N. 46
By. For verbs compounded with ‘By,’ see under ‘Bi’
Bycase, adv. == by chance. RG. 490
Byefþe. See Behoof
Byquide. See Bequest
Byȝyte. See Beget

C.
Cable, sb. RG. 148
Cacherel, sb. == catch poll. Pol. S. 151
Cage, sb. Alys. 5011
Caitiff, sb. Body and Soul, 229
Cake, sb. Cok. 55
Cales, sb. == a kind of serpent. Alys. 7094
Calf, sb. (the animal). Alys. 6351
Call, v. n. Wright’s L. P. p. 59
Call, sb. == cap worn on the head. Pol. S. 158. Fr. cale
Caluȝ, adj. == bald. Alys. 5950. AS. calo, caluw
Camel, sb. Alys. 854
Can, vb. == am able [con]. Wright’s L. P. p. 82. [cunne]. 2 s. pres.
‘cost.’ Wright’s L. P. p. 91. O. and N. 47. pret. ‘cowþe.’ RG. 29
—— == know [con]. RG. 443. [cunne]. O. and N. 48. 2 s. pres.
‘canst.’ O. and N. 560
Candle, sb. RG. 290, 561
Candlemas, sb. St Dunstan, 3
Canel, sb. == cinnamon. Wright’s L. P. p. 27. Fr canelle. Lat. canna
Cankerfret, adj. RG. 299
Canon, sb. RG. 510

Capel, sb. == horse, nag. Cok. 32. Lat. caballus
Capelclawer, sb. == horse-scrubber. Pol. S. 239
Capital, sb. (of a column). Cok. 67
Carbuncle, sb. Alys. 5252. HD. 2145
Cardinal, sb. 1280 B.
Care, sb. RG. 457
Care, v. n. == be anxious. RG. 71. Wright’s L. P. p. 54
Careful, adj. == full of care. 639 B.
Carie, sb. == carat. Alys. 6695
Carke, v. n. == pine away. Wright’s L. P. p. 54
Carol, sb. RG. 53
Carol, v. n. Alys. 196, 1045
Caronye, sb. == carcass. RG. 265
Carp, v. n. == complain. Pol. S. 149
Carpenter, sb. RG. 537
Carrion, sb. (caraing). Pol. S. 203
Cart, sb. RG. 189
Cartload, sb. HD. 895
Cartstave, sb. RG. 99
Carve, v. a. RG. 560. == cut, flay. part. ‘corven.’ Wright’s L. P. p. 35.
‘curven.’ HD. 189
Case, sb. == chance, event. RG. 528

—— == condition. Alys. 4428
Cast, v. a. RG. 511, 375
Castle, sb. RG. 371, 510; pl. ‘kasteles’ == tents. Ps. lxxvii. 28
Cat, sb. Alys. 5275
Catathleba (κατώβλεπας), == a noxious monster, mentioned in
Alys. 6564. See Pliny, H. N. viii. 32
Catch, v. a. RG. 28. pret. ‘caught.’ RG. 375. part. ‘cacchynge.’ RG.
265
Cathedral, adj. RG. 282
Caudle, sb. RG. 561
Cauldron, sb. 158 β
Caution, sb. == surety. RG. 506
—— == quarter in battle. Alys. 2811
Cavenard, sb. == villain. HD. 2389. The form ‘caynard’ is found in
Wright’s L. P. p. 110. Fr. caignard. Cotgr.
Cayre, v. a. == turn. part. ‘ycayred.’ Wright’s L. P. p. 37. AS. cerran.
Germ. kehren
Caynard. See Cavenard
Cayser, sb. == emperor. HD. 1317. Wright’s L. P. p. 32
Cayvar, adj. == hollow? Alys. 6062
Cedar, sb. Ps. ciii. 16
Cel, sb. == seal. RG. 77
Celadoyne, sb. See Celandine

Celandine, sb. == the flower. Wright’s L. P. p. 26. Lat. chelidonium.
It is the ‘ranunculus ficaria’ of botanists
Cell, sb. RG. 233
Cellar, sb. 287 B.
Cement, sb. Alys. 6177
Censer, sb. Marg. 75
Cerge, sb. == a taper. HD. 594. ON. kérti. Germ. kerze
Cert, adv. == certainly. Alys. 5803
Certain, adj. == fixed, ascertained. RG. 378, 552
Certés, adv. 898 B.
Cestred, == lodged, concealed. Ps. lxxiii. 20; cxxxviii. 12. AS.
ceaster
Chaffare, sb. == merchandise. RG. 539. AS. ceápian
Chair, sb. RG. 321
Chaisel, sb. == a woman’s upper garment. Alys. 279. ‘espéce de
vétement.’ Roq. s. v. SS. cheisil. Fr. cheinsil, v. Roq. s. v. chainse, and
Gloss. Rem. to Laȝ. iii. 502
Chalandre, sb. == goldfinch. Cok. 95
Chalcedony, sb. Cok. 92
Chalen, sb. == chill, cold. Alys. 4834
Chalice, sb. RG. 489. HD. 187
Chalktrap, sb. == pit or snare. Alys. 6070
Challenge, v. a. RG. 279, 451
Chamber, sb. 452 B.

Chamberlain. RG. 390, 490
Champion, sb. HD. 1015
Chance, sb. == condition, fortune. RG. 465
—— == chance [cheance]. RG. 210
Chancellor, sb. RG. 540, 468
Chancellory, sb. == office of chancellor. 452 B.
Chane, vb. pret. == cleft. Alys. 2228. AS. cínan. perf. cán. The ‘ch’
appears in ‘tochan,’ the pret. of ‘tocínan,’ in Laȝamon, ii. 468. Weber
wrongly derives the word from Fr. choir, and makes it mean ‘fell’
Change, sb. RG. 493
Change, vb. a. RG. 548
Chantment, sb. == enchantment. RG. 28, 149
Chapel, sb. RG. 472, 473
Chapitle, sb. == chapter of a cathedral RG. 473
Chaplain, sb. 961 B.
Chapman, sb. RG. 539
Chapter, sb. (of a cathedral). 601 B.
Char, sb. == turn, movement. Body and Soul, 79. Hence ‘ȝeynchar’
== repentance. Wright’s L. P. p. 46. SS. charren. AS. cérran, cérre.
Germ. kehren
Charge, v. a. == load. RG. 13. part. ‘icharged.’ Pol. S. 195
—— sb. == load, weight. RG. 416
—— == expense. RG. 189
Charity, sb. Pol. S. 202. ‘par charité.’ 1811 B.

Charm, sb. == spell. Alys. 81
Charming, sb. == spell. Alys. 404
Charreye, sb. == car. Alys. 5097
Charter, sb. RG. 477, 498
Chase, sb. == hunting. RG. 6
Chaste, adj. 154 B.; [cheste]. Alys. 7050. ‘chaster.’ RG. 191
Chaste, v. a. == chastise. RG. 134
Chastise, v. a. RG. 420
Chasuble, sb. == priest’s robe. 953 B. Fr. casule. Ital. casupola
Chasur, sb. == horse for hunting. Signa ante Jud. 110. Fr. chaceor
Chaterestre, sb. == a female chatterer. O. and N. 655
Chattels, sb. [chateus]. RG. 471, 569. Another form of ‘cattle’
Chattering, sb. O. and N. 744
Chaumpebataile, sb. == battle-field. Alys. 5553
Chavling, sb. == jawing. O. and N. 284, 296
Chawl, sb. == jaw. Body and Soul, 189. Pol. S. 154. AS. ceafl. SS.
chevele. pl. chæfles
Chawl, v. n. == to chide, jaw. Pol. S. 240
Cheap, sb. == haggling? Wright’s L. P. p. 39
Cheap, v. a. == buy. Pol. S. 159. AS. ceápian
Cheaping, sb. == market. Pol. S. 151
Cheek, sb. Wright’s L. P. p. 34

Cheer, sb. == comfort. 473 B.
—— == countenance. RG. 332
Cheese, sb. HD. 643
Chelde, sb. == chill, cold. Alys. 5501
Cheole, sb. == hair. M. Ode, 182. Fr. chevol
Chepe, Cheping. See Cheap, Cheaping
Chequer, sb. == chess. RG. 192; or perhaps ‘the chessboard’
Cherde, vb. pret. == turned, came. O. and N. 1656. AS. cyrran,
cérran
Chere, adj. == high? ‘the chere men of the land.’ RG. 166.
Cheson, sb. == occasion. Alys. 3930
Chess, sb. Alys. 2096
Chest, sb. == coffin. RG. 50
Cheste, sb. == strife. Alys. 29. AS. ceást
Chete, == a chewet, or pie. Wright’s L. P. p. 31
Cheui, an error for ‘cheve.’ RG. 94
Cheve, v. n. == succeed in a thing. 856 B. Fr. chevir
Chide, v. a. 2 s. pres. ‘chist.’ O. and N. 1329. AS. cídan
—— v. n. RG. 390
—— == dispute. O. and N. 287
Chief, sb. == chieftain. 1003 B.
Chief, adj. == ‘to hold in chief,’ a law term, applied to those tenants
who held their fiefs direct from the king; ‘tenants in capite.’ RG. 472

—— == principal. St Swithin, 22
Chieftain, sb. [cheventeyn]. RG. 386, 400
Chilce, sb. == childishness? M. Ode, 4. Formed from ‘child,’ as
‘milce’ from ‘mild’
Child, sb. RG. 392, 441; [chil]. O. and N. 1438
Child, v. n. == bring forth a child. Alys. 604
Childbed, sb. RG. 379
Childering, sb. == bringing forth a child. Rel. S. ii. 7
Chill, sb. RG. 7
Chimbe, sb. == cymbal. Ps. cl. 5
Chime, sb. (of bells). Alys. 1852. Dan. kime
Chin, sb. 522 β
Chinche, adj. == niggardly. HD. 1763. Fr. chice == avarice
Chirchegong, == churchgoing. RG. 380. Cf. ‘idelgong’
Chirm, sb. == chirping and screaming of birds. O. and N. 305. AS.
cyrm
Chirurgeon, sb. RG. 566
Chivalry, sb. == prowess. RG. 413
Chivauché, sb. == an expedition, a body of men. Ritson’s AS. viii.
141. Fr. chevauchée, from cheval.
Choice, sb. RG. 111
Chokering, sb. == a low chattering. O. and N. 504
Cholle, == shall. RG. 379, in the compound form ‘ycholle’

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Normal Approximation And Asymptotic Expansions Clasics In Applied Mathmatics Rabi N Bhattacharya

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