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Probability_Inequalities.pdf
Title Page
Copyright Page
Preface
Table of Contents
Chapter 1 Elementary Inequalities of Probabilities of Events
1.1 Inclusion-exclusion Formula
1.2 Corollaries of the Inclusion-exclusion Formula
1.3 Further Consequences of the Inclusion-exclusion Formula
1.4 Inequalities Related to Symmetric Difference
1.5 Inequalities Related to Independent Events
1.6 Lower Bound for Union (Chung-Erd¨os)
References
Chapter 2 Inequalities Related to Commonly Used Distributions
2.1 Inequalities Related to the Normal d.f.
2.2 Slepian Type Inequalities
2.3 Anderson Type Inequalities
2.4 Khatri-Sidak Type Inequalities
2.5 Corner Probability of Normal Vector
2.6 Normal Approximations of Binomial and Poisson Distributions
References
Chapter 3 Inequalities Related to Characteristic Functions
3.1 Inequalities Related Only with c.f.
3.2 Inequalities Related to c.f. and d.f.
3.3 Normality Approximations of c.f. of Independent Sums
References
Chapter 4 Estimates of the Difference of Two Distribution Functions
4.1 Fourier Transformation
4.2 Stein-Chen Method
4.3 Stieltjes Transformation
References
Chapter 5 Probability Inequalities of Random Variables
5.1 Inequalities Related to Two r.v.’s
5.2 Perturbation Inequality
5.3 Symmetrization Inequalities
5.4 Levy Inequality
5.5 Bickel Inequality
5.6 Upper Bounds of Tail Probabilities of Partial Sums
5.7 Lower Bounds of Tail Probabilities of Partial Sums
5.8 Tail Probabilities for Maximum Partial Sums
5.9 Tail Probabilities for Maximum Partial Sums (Continuation).
5.10 Reflection Inequality of Tail Probability (Hoff- mann- Jør-gensen)
5.11 Probability of Maximal Increment (Shao)
5.12 Mogulskii Minimal Inequality
5.13 Wilks Inequality
References
Chapter 6 Bounds of Probabilities in Terms of Moments.
6.1 Chebyshev-Markov Type Inequalities
6.2 Lower Bounds
6.3 Series of Tail Probabilities
6.4 Kolmogorov Type Inequalities
6.5 Generalization of Kolmogorov Inequality for a Submartingale
6.6 Renyi-Hajek Type Inequalities
6.7 Chernoff Inequality
6.8 Fuk and Nagaev Inequality
6.9 Burkholder Inequality
6.10 Complete Convergence of Partial Sums
References
Chapter 7 Exponential Type Estimates of Probabilities
7.1 Equivalence of Exponential Estimates
7.2 Petrov Exponential Inequalities
7.3 Hoeffding Inequality
7.4 Bennett Inequality
7.5 Bernstein Inequality
7.6 Exponential Bounds for Sums of Bounded Variables
7.7 Kolmogorov Inequalities
7.8 Prokhorov Inequality
7.9 Exponential Inequalities by Censoring
7.10 Tail Probability of Weighted Sums
References
Chapter 8 Moment Inequalities Related to One or Two Variables
8.1 Moments of Truncation
8.2 Exponential Moment of Bounded Variables
8.3 Holder Type Inequalities
8.4 Jensen Type Inequalities
8.5 Dispersion Inequality of Censored Variables
8.6 Monotonicity of Moments of Sums
8.7 Symmetrization Moment Inequatilies
8.8 Kimball Inequality
8.9 Exponential Moment of Normal Variable
8.10 Inequatilies of Nonnegative Variable
8.11 Freedman Inequality
8.12 Exponential Moment of Upper Truncated Variables
References
Chapter 9 Moment Estimates of (Maximum of) Sums of Random Variables
9.1 Elementary Inequalities
9.2 Minkowski Type Inequalities
9.3 The Case 1 < r < 2
9.4 The Case r > 2
9.5 Jack-knifed Variance
9.6 Khintchine Inequality
9.7 Marcinkiewicz-Zygmund-Burkholder Type Inequalities
9.8 Skorokhod Inequalities
9.9 Moments of Weighted Sums
9.10 Doob Crossing Inequalities
9.11 Moments of Maximal Partial Sums
9.12 Doob Inequalities
9.13 Equivalence Conditions for Moments
9.14 Serfling Inequalities
9.15 Average Fill Rate
References
Chapter 10 Inequalities Related to Mixing Sequences
10.1 Covariance Estimates for Mixing Sequences
10.2 Tail Probability on α-mixing Sequence
10.3 Estimates of 4-th Moment on ρ-mixing Sequence
10.4 Estimates of Variances of Increments of ρ-mixing Sequence
10.5 Bounds of 2 + δ-th Moments of Increments of ρ-mixing Sequence
10.6 Tail Probability on ϕ-mixing Sequence
10.7 Bounds of 2 + δ-th Moment of Increments of ϕ-mixing Sequence
10.8 Exponential Estimates of Probability on ϕ-mixing Sequence
References
Chapter 11 Inequalities Related to Associative Vari-ables
11.1 Covariance of PQD Varaibles
11.2 Probability of Quadrant on PA (NA) Sequence
11.3 Estimates of c.f.’s on LPQD (LNQD) Sequence
11.4 Maximal Partial Sums of PA Sequence
11.5 Variance of Increment of LPQD Sequence
11.6 Expectation of Convex Function of Sum of NA Sequence
11.7 Marcinkiewicz-Zygmund-Burkholder Inequality for NA Sequence
References
Chapter 12 Inequalities about Stochastic Processes and Banach Space Valued Random Variables
12.1 Probability Estimates of Supremums of a Wiener Process
12.2 Probability Estimate of Supremum of a Poisson Process
12.3 Fernique Inequality
12.4 Borell Inequality
12.5 Tail Probability of Gaussian Process
12.6 Tail Probability of Randomly Signed Independent Processes
12.7 Tail Probability of Adaptive Process
12.8 Tail Probability on Submartingale
12.9 Tail Probability of Independent Sum in B-Space
12.10 Isoperimetric Inequalities
12.11 Ehrhard Inequality
12.12 Tail Probability of Normal Variable in B-Space
12.13 Gaussian Measure on Symmetric Convex Sets
12.14 Equivalence of Moments of B-Gaussian Variables
12.15 Contraction Principle
12.16 Symmetrization Inequalities in B-Space
12.17 Decoupling Inequality
References
Zhengyan Lin
Zhidong Bai
Probability Inequalities
Zhengyan Lin
Zhidong Bai
Probability Inequalities
Zhidong Bai
School of Mathematics and Statistics
Northeast Normal University, China
baizd@nenu.edu.cn
Department of Statistics and Applied Probabilty
National University of Singapore, Singapore
stabaizd@leonis.nus.edu.sg
Authors
Zhengyan Lin
Department of Mathematics
Zhejiang University, China
zlin@zju.edu.cn
ISBN 978-7-03-025562-4
Science Press Beijing
ISBN 978-3-642-05260-6 e-ISBN 978-3-642-05261-3
Springer Heidelberg Dordrecht London New York
Library of Congress Control Number: 2009938102
© Science Press Beijing and Springer-Verlag Berlin Heidelberg 2010
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concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,
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Preface
In almost every branch of quantitative sciences, inequalities play an im-
portant role in its development and are regarded to be even more impor-
tant than equalities. This is indeed the case in probability and statis-
tics. For example, the Chebyshev, Schwarz and Jensen inequalities are
frequently used in probability theory, the Cramer-Rao inequality plays
a fundamental role in mathematical statistics. Choosing or establishing
an appropriate inequality is usually a key breakthrough in the solution
of a problem, e.g. the Berry-Esseen inequality opens a way to evaluate
the convergence rate of the normal approximation.
Research beginners usually face two difficulties when they start resear-
ching—they choose an appropriate inequality and/or cite an exact ref-
erence.
In literature, almost no authors give references for frequently
used inequalities, such as the Jensen inequality, Schwarz inequality, Fa-
tou Lemma, etc. Another annoyance for beginners is that an inequality
may have many different names and reference sources. For example,
the Schwarz inequality is also called the Cauchy, Cauchy-Schwarz or
Minkovski-Bnyakovski inequality. Bennet, Hoeffding and Bernstein in-
equalities have a very close relationship and format, and in literature
some authors cross-cite in their use of the inequalities. This may be due
to one author using an inequality and subsequent authors just simply
copying the inequality’s format and its reference without checking the
original reference. All this may distress beginners very much.
The aim of this book is to help beginners with these problems. We
provide a place to find the most frequently used inequalities, their proofs
(if not too lengthy) and some references. Of course, for some of the more
popularly known inequalities, such as Jensen and Schwarz, there is no
necessity to give a reference and we will not do so.
The wording “frequently used” is based on our own understanding.
It can be expected that many important probability inequalities are not
ii
Preface
collected in this work. Any comments and suggestions will be appreci-
ated.
The writing of the book is supported partly by the National Science
Foundation of China.
The authors would like to express their thanks to Ron Lim Beng Seng
for improving our English in this book.
Zhengyan Lin
May, 2009
Contents
Chapter 1 Elementary Inequalities of Probabilities of
Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Inclusion-exclusion Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Corollaries of the Inclusion-exclusion Formula . . . . . . . . . . . . . . . 2
1.3 Further Consequences of the Inclusion-exclusion Formula . . . . 2
1.4 Inequalities Related to Symmetric Difference . . . . . . . . . . . . . . . . 6
1.5 Inequalities Related to Independent Events . . . . . . . . . . . . . . . . . . 6
1.6 Lower Bound for Union (Chung-Erd¨os) . . . . . . . . . . . . . . . . . . . . . . 8
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8
Chapter 2 Inequalities Related to Commonly Used Dis-
tributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1 Inequalities Related to the Normal d.f. . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Slepian Type Inequalities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12
2.3 Anderson Type Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4 Khatri-ˇSid´ak Type Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.5 Corner Probability of Normal Vector . . . . . . . . . . . . . . . . . . . . . . . 19
2.6 Normal Approximations of Binomial and Poisson Distri-
butions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Chapter 3 Inequalities Related to Characteristic Fun-
ctions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23
3.1 Inequalities Related Only with c.f.. . . . . . . . . . . . . . . . . . . . . . . . . .23
3.2 Inequalities Related to c.f. and d.f. . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3 Normality Approximations of c.f. of Independent Sums . . . . . 27
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Chapter 4 Estimates of the Difference of Two Distri-
bution Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.1 Fourier Transformation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29
4.2 Stein-Chen Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.3 Stieltjes Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
iv
Contents
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Chapter 5 Probability Inequalities of Random Vari-
ables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .37
5.1 Inequalities Related to Two r.v.’s. . . . . . . . . . . . . . . . . . . . . . . . . . .37
5.2 Perturbation Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.3 Symmetrization Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.4 L´evy Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.5 Bickel Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.6 Upper Bounds of Tail Probabilities of Partial Sums. . . . . . . . .44
5.7 Lower Bounds of Tail Probabilities of Partial Sums . . . . . . . . . 44
5.8 Tail Probabilities for Maximum Partial Sums. . . . . . . . . . . . . . . 45
5.9 Tail Probabilities for Maximum Partial Sums (Contin-
uation). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .46
5.10 Reflection Inequality of Tail Probability (Hoffmann-
Jørgensen) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.11 Probability of Maximal Increment (Shao) . . . . . . . . . . . . . . . . . 48
5.12 Mogulskii Minimal Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.13 Wilks Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Chapter 6 Bounds of Probabilities in Terms of Mo-
ments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .51
6.1 Chebyshev-Markov Type Inequalities . . . . . . . . . . . . . . . . . . . . . . . 51
6.2 Lower Bounds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .52
6.3 Series of Tail Probabilities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .53
6.4 Kolmogorov Type Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
6.5 Generalization of Kolmogorov Inequality for a Submar-
tingale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
6.6 R´enyi-H´ajek Type Inequalities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .57
6.7 Chernoff Inequality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .60
6.8 Fuk and Nagaev Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
6.9 Burkholder Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
6.10 Complete Convergence of Partial Sums . . . . . . . . . . . . . . . . . . . . 65
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
Chapter 7 Exponential Type Estimates of Probabilities . . . 67
7.1 Equivalence of Exponential Estimates . . . . . . . . . . . . . . . . . . . . . . 67
7.2 Petrov Exponential Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
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