Inequalities in Analysis and Probability

306 Pages·2017·3.977 MB·english

by  Odile Pons

INEQUALITIES
IN
ANALYSIS
AND
PROBABILITY
Second Edition
Odile Pons
National Institute for Agronomical Research, France

World Scientific
10139hc_9789813143982_tp.indd 2 13/4/16 9:07 AM
Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Cataloging-in-Publication Data
Names: Pons, Odile.
Title: Inequalities in analysis and probability / by Odile Pons
(French National Institute for Agronomical Research, France).
Description: Second edition. | New Jersey : World Scientific, 2016. |
Includes bibliographical references and index.
Identifiers: LCCN 2016038435 | ISBN 9789813143982 (hardcover)
Subjects: LCSH: Inequalities (Mathematics)
Classification: LCC QA295 .P66 2016 | DDC 512.9/7--dc23
LC record available at https://lccn.loc.gov/2016038435
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Copyright © 2017 by World Scientific Publishing Co. Pte. Ltd.
Printed in Singapore
LaiFun - Inequalities in Analysis and Probability.indd 1 18-10-16 4:18:22 PM
August17,2016 14:54 ws-book9x6 InequalitiesinAnalysisandProbability 10139-main pagev
Preface
The most important changes made in this edition are the insertion of
two chapters: Chapter 5 on stochatic calculus with first-order differentia-
tion and exponential (sub)-martingales,and Chapter 7 on time-continuous
Markov processes, the renewal equations and the Laplace transform of the
processes. I have added to Chapter 4 a section on the p-order variations
of a process and modified the notation to extend the inequalities to local
martingales with a discontinuous process ofquadratic variations,the nota-
tions of the previousedition werenotmodified for acontinuous martingale
and for a point process with a continuous predictable compensator. I have
also corrected a few misprints and errors.
ExamplesofPoissonandGaussianprocessesillustratethetextandtheir
investigation leads to general results for processes with independent incre-
ments and for semi-martingales.
Odile M.-T. Pons
February 2016
Preface of the First Edition

TheinequalitiesinvectorspacesandfunctionalHilbertspacesarenaturally
transposed to random variables, martingales and time indexed stochastic
processes with values in Banachspaces. The inequalities for transforms by
convex functions are examples of the diffusion of simple arithmetic results
to a wide range of domains in mathematics. New inequalities have been
developedindependently inthese fields. Thisbookaimsto giveanaccount
ofinequalitiesinanalysisandprobabilityandtocompleteandextendthem.
The introduction gives a survey of classicalinequalities in severalfields
withthemainideasoftheirproofsandapplicationsoftheanalyticinequal-
August17,2016 14:54 ws-book9x6 InequalitiesinAnalysisandProbability 10139-main pagevi
ities to probability. This is not an exhaustive list. They are compared
and sometimes improved with simple proofs. Further developments in the
literaturearementioned. Thebookisorganizedaccordingtothemaincon-
cepts and it provides new inequalities for sums of random variables, their
maximum, martingales, Brownian motions and diffusion processes, point
processes and their suprema.
The emphasis on the inequalities is aimed at graduate students and re-
searchershavingthebasicknowledgeofcoursesinAnalysisandProbability.
The concepts of integration theory and of probabilities are supposed to be
known, so the fundamental inequalities in these domains are acquired and
references to other publications are added to complete the topic whenever
possible. The book contains many proofs, in particular basic inequalities
for martingales with discrete or continuous parameters in detail and the
progress in severaldirections are easily accessible to the readers. They are
illustrated by applications in probability.
I undertook this work in order to simplify the approach of uniform
bounds for stochastic processes in functional classes. In the statistical ap-
plications, the assumptions for most results of this kind are specific to
another distance than the uniform distance. Here, the results use inequal-
ities of Chapter 4 between the moments of martingales and those of their
predictable variations,then the conditions and the constants of the proba-
bilisticbounddifferfromthoseoftheotherauthors. Duringthepreparation
ofthebook,Iaddedotherinequalitieswhilereadingpapersandbookscon-
taining errors and unproved assertions; it should therefore fill some gaps.
It does not cover the convex optimization problems and the properties of
their solutions. It can be used as an introduction to more specific domains
of the functional analysis or probability theory and as a reference for new
applications to the asymptotic behaviour of non-standard empirical pro-
cesses in statistics. Several applications to the tail behaviour of processes
are developed in the following chapters.

Odile M.-T. Pons
April 2012
August17,2016 14:54 ws-book9x6 InequalitiesinAnalysisandProbability 10139-main pagevii
Contents
Preface v
1. Preliminaries 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Cauchy and Ho¨lder inequalities . . . . . . . . . . . . . . . 2
1.3 Inequalities for transformed series and functions . . . . . . 6
1.4 Applications in probability . . . . . . . . . . . . . . . . . . 9
1.5 Hardy’s inequality . . . . . . . . . . . . . . . . . . . . . . 13
1.6 Inequalities for discrete martingales . . . . . . . . . . . . . 15
1.7 Martingales indexed by continuous parameters . . . . . . 20
1.8 Large deviations and exponential inequalities . . . . . . . 24
1.9 Functional inequalities . . . . . . . . . . . . . . . . . . . . 28
1.10 Content of the book . . . . . . . . . . . . . . . . . . . . . 30
2. Inequalities for Means and Integrals 33
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2 Inequalities for means in real vector spaces. . . . . . . . . 33
2.3 Ho¨lder and Hilbert inequalities . . . . . . . . . . . . . . . 38
2.4 Generalizations of Hardy’s inequality . . . . . . . . . . . . 40
2.5 Carleman’s inequality and generalizations . . . . . . . . . 49
2.6 Minkowski’s inequality and generalizations . . . . . . . . . 50
2.7 Inequalities for the Laplace transform . . . . . . . . . . . 54
2.8 Inequalities for multivariate functions . . . . . . . . . . . . 57
3. Analytic Inequalities 63
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 63
August17,2016 14:54 ws-book9x6 InequalitiesinAnalysisandProbability 10139-main pageviii
3.2 Bounds for series . . . . . . . . . . . . . . . . . . . . . . . 65
3.3 Cauchy’s inequalities and convex mappings . . . . . . . . 68
3.4 Inequalities for the mode and the median . . . . . . . . . 72
3.5 Mean residual time . . . . . . . . . . . . . . . . . . . . . . 77
3.6 Functional equations . . . . . . . . . . . . . . . . . . . . . 79
3.7 Carlson’s inequality . . . . . . . . . . . . . . . . . . . . . . 84
3.8 Functional means . . . . . . . . . . . . . . . . . . . . . . . 87
3.9 Young’s inequalities. . . . . . . . . . . . . . . . . . . . . . 90
3.10 Entropy and information . . . . . . . . . . . . . . . . . . . 93
4. Inequalities for Martingales 99
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.2 Inequalities for sums of independent random variables . . 100
4.3 Inequalities for discrete martingales . . . . . . . . . . . . . 107
4.4 Inequalities for the maximum . . . . . . . . . . . . . . . . 112
4.5 Inequalities for martingales indexed by R . . . . . . . . . 113
+
4.6 Inequalities for p-order variations . . . . . . . . . . . . . . 118
4.7 Poissonprocesses . . . . . . . . . . . . . . . . . . . . . . . 122
4.8 Brownian motion . . . . . . . . . . . . . . . . . . . . . . . 126
4.9 Diffusion processes . . . . . . . . . . . . . . . . . . . . . . 131
4.10 Martingales in the plane . . . . . . . . . . . . . . . . . . . 134
5. Stochastic Calculus 137
5.1 Stochastic integration . . . . . . . . . . . . . . . . . . . . 137
5.2 Exponential solutions of differential equations . . . . . . . 139
5.3 Exponential martingales, submartingales . . . . . . . . . . 141
5.4 Gaussian processes . . . . . . . . . . . . . . . . . . . . . . 145
5.5 Processes with independent increments . . . . . . . . . . . 149
5.6 Semi-martingales . . . . . . . . . . . . . . . . . . . . . . . 153
5.7 Level crossing probabilities. . . . . . . . . . . . . . . . . . 154
5.8 Sojourn times . . . . . . . . . . . . . . . . . . . . . . . . . 159
6. Functional Inequalities 163
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 163
6.2 Exponential inequalities for functional empirical
processes. . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
6.3 Exponential inequalities for functional martingales . . . . 171
6.4 Weak convergence of functional processes . . . . . . . . . 175
August17,2016 14:54 ws-book9x6 InequalitiesinAnalysisandProbability 10139-main pageix
6.5 Differentiable functionals of empirical processes . . . . . . 178
6.6 Regression functions and biased length . . . . . . . . . . . 182
6.7 Regression functions for processes . . . . . . . . . . . . . . 187
6.8 Functional inequalities and applications . . . . . . . . . . 188
7. Markov Processes 191
7.1 Ergodic theorems . . . . . . . . . . . . . . . . . . . . . . . 191
7.2 Inequalities for Markov processes . . . . . . . . . . . . . . 195
7.3 Convergence of diffusion processes . . . . . . . . . . . . . 196
7.4 Branching process . . . . . . . . . . . . . . . . . . . . . . 197
7.5 Renewal processes . . . . . . . . . . . . . . . . . . . . . . 201
7.6 Maximum variables . . . . . . . . . . . . . . . . . . . . . . 207
7.7 Shock process . . . . . . . . . . . . . . . . . . . . . . . . . 212
7.8 Laplace transform. . . . . . . . . . . . . . . . . . . . . . . 217
7.9 Time-space Markov processes . . . . . . . . . . . . . . . . 222
8. Inequalities for Processes 227
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 227
8.2 Stationary processes . . . . . . . . . . . . . . . . . . . . . 228
8.3 Ruin models . . . . . . . . . . . . . . . . . . . . . . . . . . 230
8.4 Comparison of models . . . . . . . . . . . . . . . . . . . . 236
8.5 Moments of the processes at T . . . . . . . . . . . . . . . 238
a
8.6 Empirical process in mixture distributions . . . . . . . . . 240
8.7 Integral inequalities in the plane . . . . . . . . . . . . . . 243
8.8 Spatial point processes . . . . . . . . . . . . . . . . . . . . 245
9. Inequalities in Complex Spaces 253
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 253
9.2 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 256
9.3 Fourier and Hermite transforms . . . . . . . . . . . . . . . 260
9.4 Inequalities for the transforms . . . . . . . . . . . . . . . . 266
9.5 Inequalities in C . . . . . . . . . . . . . . . . . . . . . . . 268
9.6 Complex spaces of higher dimensions . . . . . . . . . . . . 269
9.7 Stochastic integrals . . . . . . . . . . . . . . . . . . . . . . 273
Appendix A Probability 277
A.1 Definitions and convergences in probability spaces. . . . . 277
A.2 Boundary-crossingprobabilities . . . . . . . . . . . . . . . 282
August17,2016 14:54 ws-book9x6 InequalitiesinAnalysisandProbability 10139-main pagex
A.3 Distances between probabilities . . . . . . . . . . . . . . . 283
A.4 Expansions in L2(R) . . . . . . . . . . . . . . . . . . . . . 286
Bibliography 289
Index 295
August17,2016 14:54 ws-book9x6 InequalitiesinAnalysisandProbability 10139-main page1
Chapter 1
Preliminaries
1.1 Introduction
The originofthe inequalitiesfor convexfunctions is the inequalities inreal
vector spaces which have been extended to functional spaces by limits in
Lebesgue integrals. They are generalized to inequalities for the tail distri-
butionofsumsofindependentordependentvariables,underconditionsfor
the convergence of their variance, and to inequalities for the distribution
of martingales indexed by discrete or continuous sets. These inequalities
are the decisive arguments for bounding series, integrals or moments of
transformed variables and for proving other inequalities.
Theconvergencerateofsumsofvariableswithmeanzeroisdetermined
by probability inequalities which prove that a sum of variables normalized
by the exact convergence rate satisfies a compactness property. If the nor-
malization has a smaller order than its convergence rate, the upper bound
oftheinequalityisoneandittendstozeroiftheorderofthenormalization
is larger.
Many probability results are related to the Laplace transform, such
as Chernoff’s large deviations theorem, Bennett’s inequalities and other

exponential inequalities for sums of independent variables. This subject
has been widely explored since the review papers of the Sixth Berkeley
SymposiuminMathematicalStatisticsandProbability(1972)whichcovers
many inequalities for martingales,Gaussianand Markovprocessesand the
related passage problems and sojourn times. Some of them are revisited
andextendedafterabriefreviewinthischapter. Theupperboundsforthe
tail probability of the maximum of n variables depend on n, in the same
way, the tail probability of the supremum of functional sums have upper
bounds depending on the dimension of the functional classes.
1
August17,2016 14:54 ws-book9x6 InequalitiesinAnalysisandProbability 10139-main page2
2 Inequalities inAnalysis and Probability
1.2 Cauchy and H¨older inequalities
Inequalities for finite series were first written as inequalities in a vector
space V provided with an Euclidean norm (cid:2)x(cid:2), for x in V. The scalar
product of x and y in V is defined from the norm by
1
<x,y >= {(cid:2)x+y(cid:2)2−(cid:2)x(cid:2)2−(cid:2)y(cid:2)2} (1.1)
2
and,conversely,anEuclideannormisa(cid:2) -normrelatedto thescalarprod-
2
uct as
(cid:2)x(cid:2)=<x,x>21 .
From the definition (1.1), the norms of vectors x and y of an Euclidean
vector space V satisfy the geometric equalities
(cid:2)x+y(cid:2)2+(cid:2)x−y(cid:2)2 =2((cid:2)x(cid:2)2+(cid:2)y(cid:2)2) (1.2)
(cid:2)x+y(cid:2)2−(cid:2)x−y(cid:2)2 =4<x,y >.
The space (cid:2) (V) is the space of series of V with a finite Euclidean norm.
2
An orthonormalbasis(ei)1≤i ofV is defined bythe orthogonalityproperty
< e ,e >= 0 for i (cid:3)= j and by the normalization (cid:2)e (cid:2) = 1 for i ≥ 1. Let
i j i
V bea vectorspaceofdimensionn,forexampleV =Rn, foranintegern
n n
and V∞ be its limit as n tends to infinity. Every vector x of (cid:2)2(Vn), n≥1,
is the sum of its projections in the orthonormal basis
(cid:2)n
x= <x,e >e ,
i i
i=1
its coordonatesinthe basis arex =<x,e >,i=1,...,n, andits normis
(cid:3) i i
(cid:2)x(cid:2)(cid:3)2 =( ni=1x2i)21. In(cid:2)2(V∞), avectorxisthelimitasn(cid:3)tendstoinfinity
of ni=1 < x,ei > ei and its norm is the finite limit of ( ni=1x2i)12 as n
tends to infinity. The space (cid:2) (V ), 1 ≤ p < ∞, is defined with respect to
p n
the norm
(cid:2)n
(cid:2)x(cid:2)p =( |xi|p)p1
i=1
and the space (cid:2)∞(Vn) is the space of vector with a finite uniform norm
(cid:2)x(cid:2)∞ = max1≤i≤n|xi|. In (cid:2)p(V∞) and (cid:2)∞(V∞), the norms are defined
as the limits of the norms of (cid:2) (V ) as n tend to infinity. The norms
p n
((cid:2)x(cid:2)p)0<p≤∞ are an increasing sequence for every x in a vector space V.
The triangular inequality is (cid:2)x + y(cid:2) ≤ (cid:2)x(cid:2) + (cid:2)y(cid:2) with equality if and
only if < x,y >= 0. Consequently, for all x and y in a vector space
|(cid:2)x(cid:2)−(cid:2)y(cid:2)|≤(cid:2)x−y(cid:2).