本书尽可能完整地收集整理概率统计中的常用不等式，包括有关事件的概率的初等不等式，关于常用分布的不等式，关于特征函数的不等式，两个分布函数的差的估计，随机变量的概率不等式，用矩估计概率的界，概率的指数型估计，关于一个或两个随机变量的矩不等式，随机变量和的(极大的)矩估计，关于相依随机变量的不等式，关于随机过程和取值于Banach空间的随机变量的不等式等。

Inequality has become an essential tool in many areas of mathematical research, for example in probability and statistics where it is frequently used in the proofs. Probability Inequalities covers inequalities related with events, distribution functions, characteristic functions, moments and random variables (elements) and their sum. The book shall serve as a useful tool and reference for scientists in the areas of probability and statistics, and applied mathematics.

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-ErdSs)

 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 (HoffmannJorgensen)

 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 HSlder 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 Serfiing 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 p-mixing Sequence

 10.4 Estimates of Variances of Increments of p-mixing Sequence

 10.5 Bounds of 2+δ-th Moments of Increments of p-mixing Sequence

 10.6 Tail Probability on g-mixing Sequence

 10.7 Bounds of 2+δ-th Moment of Increments of mixing Sequence

 10.8 Exponential Estimates of Probability on mixing Sequence

 References

Chapter 11 Inequalities Related to Associative Variables

 11.1 Covariance of PQD Varalbles

 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 DecoupIing Inequality

 References