Martingale Limit Theory and Its Application

309 Pages·1980·11.968 MB·English

by  Peter Hall, C. C. Heyde

Thisisavolumein
PROBABILITY AND MATHEMATICAL STATISTICS
ASeriesofMonographsandTextbooks
Editors: Z. W.Birnbaum and E. Lukacs
Acompletelistof titlesinthisseriescan beobtainedfrom thePublisherupon request.
MartIngale
LImIt Theory
and
It. ApplIcatIon
P. Hall
Department ofStatistics, SGS
Australian National University
Canberra,Australia
C. C. Heyde
CSIRODivisionofMathematics and Statistics
CanberraCity, Australia
1980
ACADEMIC PRESS
A Subsidiary ofHarcourt BraceJovanovich, Publishers
NewYork London Toronto Sydney San Francisco
The history ofprobability (and ofmathematics in general) shows a
stimulating interplay ofthe theory and applications: theoreticalprogress
opens new fields ofapplications, and in turn applications lead to new
problems andfruitfulresearch. Thetheoryofprobabilityisnowappliedto
many diversefields, and theflexibility ofageneraltheory isrequired to
provide appropriatetoolsfor sogreatavarietyofneeds.
w.Feller
...theepistemologicalvalueofthetheoryofprobabilityisrevealedonlyby
limit theorems. Moreover, without limit theorems it is impossible to
understand therealcontent oftheprimaryconceptofalloursciences-the
concept ofprobability.
B.V.Gnedenkoand A. N. Kolmogorov
COPYRIGHT© 1980, BYACADEMIC PRESS,INC.
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LibraryofCongressCatalogingin PublicationData
Hall,P
Martingalelimit theoryand itsapplication.
(Probabilityand mathematicalstatistics)
Bibliography: p.
Includes indexes.
1. Martingales(Mathematics) 2. Limit
theorems(Probabilitytheory) I. Heyde, C.C.,
jointauthor. II. Title.
QA274.5.H34 519.2'87 80-536
ISBN 0-12-319350-8
PRINTEDIN THEUNITED STATESOF AMERICA
80 81 82 83 987654321
The history ofprobability (and ofmathematics in general) shows a
stimulating interplay ofthe theory and applications: theoreticalprogress
opens new fields ofapplications, and in turn applications lead to new
problems andfruitfulresearch. Thetheoryofprobabilityisnowappliedto
many diversefields, and theflexibility ofageneraltheory isrequired to
provide appropriatetoolsfor sogreatavarietyofneeds.
w.Feller
...theepistemologicalvalueofthetheoryofprobabilityisrevealedonlyby
limit theorems. Moreover, without limit theorems it is impossible to
understand therealcontent oftheprimaryconceptofalloursciences-the
concept ofprobability.
B.V.Gnedenkoand A. N. Kolmogorov
COPYRIGHT© 1980, BYACADEMIC PRESS,INC.
ALLRIGHTSRESERVED.
NO PART OFTHISPUBLICATIONMAYBEREPRODUCEDOR
TRANSMITTEDIN ANYFORMORBYANY MEANS,ELECTRONIC
OR MECHANICAL, INCLUDINGPHOTOCOPY, RECORDING, ORANY
INFORMATION STORAGE ANDRETRIEVAL SYSTEM,WITHOUT
PERMISSIONIN WRITINGFROM THEPUBLISHER.
ACADEMIC PRESS, INC.
111FifthAvenue,NewYork,NewYork10003
United Kingdom Edition publishedby
ACADEMIC PRESS, INC. (LONDON) LTD.
24/28OvalRoad,LondonNW1 7DX
LibraryofCongressCatalogingin PublicationData
Hall,P
Martingalelimit theoryand itsapplication.
(Probabilityand mathematicalstatistics)
Bibliography: p.
Includes indexes.
1. Martingales(Mathematics) 2. Limit
theorems(Probabilitytheory) I. Heyde, C.C.,
jointauthor. II. Title.
QA274.5.H34 519.2'87 80-536
ISBN 0-12-319350-8
PRINTEDIN THEUNITED STATESOF AMERICA
80 81 82 83 987654321
Preface
This bookwascommenced byoneoftheauthorsinlate1973inresponseto
agrowingconvictionthattheasymptoticpropertiesofmartingalesprovidea
keyprototype ofprobabilistic behaviour,whichisofwideapplicability. The
evidence in favor of such a proposition has beenamassing rapidly over the
interveningyears-so rapidly indeed thatthe subject keptescapingfrom the
confines of the text. The coauthorjoined the project in late 1977.
Thethesisofthisbook,thatmartingalelimittheoryoccupiesacentralplace
in probability theory, may still be regarded as controversial. Certainly the
story is far from complete on the theoretical side, and many interesting
questions remainoversuchissuesastherelationshipbetweenmartingalesand
processes embeddable in or approximable by Brownian motion.' On the
other hand, the picture ismuch clearer on the applied side.The vitality and
principal source ofinspiration ofprobabilitytheory comesfrom itsapplica
tions. The mathematical modeling of physical reality and the inherent
nondeterminism of many systems provide an expanding domain of rich
pickings in which martingale limit results are demonstrably of great
usefulness.
The effectivebirth of probability as a subject took place in hardly more
thana decadearound 1650,2and ithas beenlargelyweddedtoindependence
theory for some 300 years. For all the intrinsic importance and intuitive
content of independence, it isnot a vital requirementfor the three keylimit
laws of probability-the strong law of large numbers, the central limit
theorem, and the law of the iterated logarithm. As far as these results are
concerned, the time has come to move to a more general and flexible
framework inwhichsuitablegeneralizationscanbeobtained.Thisisthestory
ofthefirst partofthe book, inwhichitisargued thatmartingalelimittheory
provides the most general contemporary setting for the key limit trio. The
basicmartingaletools,particularlytheinequalities, haveapplicationsbeyond
'Forrecentcontributions,seeDrogin (1973),Philipp andStout(1975),and Monroe(1978).
Wehavenot directlyconcerned ourselveswiththeseissues.
2See,for example, Heyde and Seneta (1977,Chapter I).
Ix
x PREFACE
the realm of limit theory. Moreover, extensions of the martingale concept
offer the prospect of increased scope for the methodology.!
Historically, thefirstmartingalelimittheorems weremotivatedbyadesire
to extend the theory for sums of independent randomvariables. Verylittle
attentionwaspaid topossibleapplications,and itisonlyinmuchmorerecent
times that applied probability and mathematical statistics have been a real
force behind the development of martingale theory.! The independence
theory has proved inadequate for handling contemporary developments in
manyfields.Independence-basedresultsarenotavailableformanystochastic
systems, and in many more an underlying regenerative behaviour must be
found in orderto employthem. Onthe otherhand, relevantmartingalescan
almost always be constructed, for example by devicessuch as centering by
subtracting conditional expectations given the past and then summing.
In this book wehave chosentoconfineour attentiontodiscretetime.The
basic martingale limit results presented here can be expected to have
corresponding versions in continuous time, but the context has too many
quite different ramifications and connotations!to betreatedsatisfactorilyin
parallel with the case of discrete time.
The word application rather than applications in the title of the book
reflectsthe scopeoftheexamplesthatarediscussed.The rapidly burgeoning
list ofapplications has rendered futile any attempt at an exhaustive or even
comprehensive treatment within the confines of a singlemonograph. As a
sample of the very recent diversity which wedo not treat, wemention the
couponcollectors problem[Sen(1979)],randomwalkswithrepulsion[Pakes
(1980)], the assessment of epidemics [Watson (1980a, b)], the weak con
vergence of U-statistics [Loynes (1978), Hall (1979)]and of the empirical
process [Loynes (1978)]and ofthe log-likelihood process[Halland Loynes
(1977)],and determiningtheorderofanautoregression[Hannanand Quinn
(1979)].Martingale methods havealsofound applicationinmany areas that
are not usually associated with probability or statistics. For example,
martingaleshavebeenusedasadescriptivedeviceinmathematicaleconomics
for over ten years, and more recently the limit theory has proved to be a
powerful tool.6Ourapplicationsratherreflecttheauthors'interests,although
it is hoped that they are diverse enough to establish beyond any doubt the
usefulness of the methodology.
The book isintended for useasareferencework ratherthanasatextbook,
although itshould besuitable for certainadvanced coursesorseminars.The
3For example linear martingales [McQueen (1973)], weak martingales [Nelson (1970),
Berman(1976)],and mixingales[McLeish(1975b,1977)].
4Thisisalittleironicinviewofthe roots ofthe martingale concept in gambling theory.
sFor example, throughtheassociation withstochastic integrals.
6See,for example, Foldes(1978),Plosser etal.(1979),and Pagan (1979).
PREFACE xl
prerequisite is a sound basic course in measure theoretic probability, and
beyond that the treatment is essentially self-contained.
In bringing this book to itsfinal form wehavereceivedadvicefrom many
people.Ourgrateful thanksaredueparticularlytoG.K.Eaglesonandalsoto
D. Aldous, E.J. Hannan,M. P. Quine, G.E.H. Reuter, H. Rootzen.andD,
J. Scott, who have suggested corrections and other improvements. Thanks
are alsodueto thevarious typistswhostruggledwiththemanuscriptand last
but not least to our wivesfor their forbearance.
C. C. HEYDE
P. HALL
Canberra, Australia
November 1979
Notation
The following notation is used throughout the book.
a.s. almost surely (that is, with probability one)
Li.d. independent and identically distributed
p.gJ. probability generating function
r.v. random variable
CLT central limit theorem
LIL law ofthe iterated logarithm
SLLN strong law oflarge numbers
ML maximum likelihood.
Almost sureconvergence, convergence in probability,andconvergencein
distribution are denoted by~',!., and· ~, respectively.
Fora randomvariable X, we use II X lipfor(EIXIP)I/P,p >0,whilevarX
denotes the variance ofX.
Themetricspace qo,I]isthespaceofcontinuousfunctionsontheinterval
[0,1] with the uniform metric p defined by
p(x,y) =sup Ix(t) - y(tH
OS,Sf
ThecomplementofaneventEisdenoted byIt, andtheindicatorfunction
of E by [(E), where
ifwEE
otherwise.
The normal distribution with mean p. and variance 02 is denoted by
N( p., 02).
The realand imaginaryparts ofafunctionfaredenoted byRefand1m/.
respectively.
Forrealnumbers, x+denotes max(0,x), andsgnxisthesignofx,whilea
/\ bismin(a,b).
The transpose ofa vector vis denoted byv',andthetraceofamatrixA is
written as tr A.
The square root ofa nonnegative variable is taken to be nonnegative.
xii
1
Introduction
1.1. General Definition
Let (Ω,J^P) be a probability space: Ω is a set, ^ a σ-field of subsets of Ω,
and Ρ a probability measure defined on ^. Let / be any interval of the form
(fl,b), (α,ί?] or [α,ί?] of the ordered set {-oo,... ,-1,0,1, ,00}.
Let {J^,, η 6 /} be an increasing sequence of σ-fields of sets. Suppose that
{Z„, η G /} is a sequence of random variables on Ω satisfying
(i) Z„ is measurable with respect to
(ii) E\Z„\ < O),
(iii) E{Z„\^J = a.s. for all ηκη,πι,πΕί.
Then, the sequence {Z„,nG 1} is said to be a martingale with respect to
{^„,nel}. We write that {Z„,^„,nGl} is a martingale. If (i) and (ii)
are retained and (iii) is replaced by the inequality E{Z„\^J ^ Z^ a.s.
{E{Z„\^J ^ Z^ a.s.), then {Z„,#„, ne 1} is called a submartingale (super-
martingale).
A reverse martingale or backwards martingale {Z„, η G /} is defined with
respect to a decreasing sequence of σ-fields {^„, η G /}. It satisfies conditions
(i) and (ii) above, and instead of (iii),
(iii') E{Z„\^J = a.s. for all m > n, n, m G /.
Clearly {Ζ^,^^, 1 ^i^n} is a reverse martingale if and only if {Z„_i+i,
^„-i+i,l ^i ^n} is a martingale, and so the theory for finite reverse
martingales is just a dual of the theory for finite (forward) martingales. The
duality does not always extend so easily to limit theory.
1.2. Historical Interlude
The name martingale was introduced into the modern probabilistic litera­
ture by Ville (1939) and the subject brought to prominence through the work
of Doob in the 1940s and early 1950s.
2 1. INTRODUCTION
Martingale theory, like probability theory itself, has its origins partly in
gambling theory, and the idea of a martingale expresses a concept of a fair
game (Z,, can represent the fortune of the gambler after η games and the
information contained in the first η games). The term martingale has, in fact,
a long history in a gambling context, where originally it meant a system for
recouping losses by doubling the stake after each loss. The Oxford English
Dictionary dates this usage back to 1815. The modern concept dates back at
least to a passing reference in Bachelier (1900).
Work on martingale theory by Bernstein (1927,1939,1940,1941) and Levy
(1935a,b, 1937) predates the use of the name martingale. These authors in­
troduced the martingale in the form of consecutive sums with a view to
generalizing limit results for sums of independent random variables. The
subsequent work of Doob however, including the discovery of the celebrated
martingale convergence theorem, completely changed the direction of the
subject. His book (1953) has remained a major influence for nearly three
decades. It is only comparatively recently that there has been a resurgence
of real interest and activity in the area of martingale limit theory which deals
with generalizations of results for sums of independent random variables.
It is with this area that our book is primarily concerned.
1.3. The Martingale Convergence Theorem
This powerful result has provided much motivation for the continued
study of martingales.
Theorem. Let {Z„,J^„, n>l} be an L^-bounded submartingale. Then
there exists a random variable Ζ such that lim„_ooZ„ = Ζ a,s. and E\Z\ ^
liminf„^oo £|Z„| < QO. // the submartingale is uniformly integrable, then Z„
converges to Ζ in L\ and if {Z„,^„} is an L^-bounded martingale, then Z„
converges to Ζ in L^.
This is an existence theorem; it tells us nothing about the limit random
variable save that it has a finite first or second moment. The theorem seems
rather unexpected a priori and it is a powerful tool which has led to a number
of interesting results for which it seems essentially a unique method of
approach. Of course one is often still faced with finding the limit law, but
that can usually be accomplished by other methods.
As a simple example of the power of the theorem, consider its application
to show that if = Yj=i is a sum of independent random variables with
converging in distribution as π oo, then S„ converges a.s. This result is
a straightforward consequence of the martingale convergence theorem when