概率不等式
1970-1
科学出版社
Zhengyan Lin, Zhidong Bai
181
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 arefrequently used in probability theory， the Cramer-Rao inequality playsa fundamental role in mathematical statistics. Choosing or establishingan appropriate inequality is usually a key breakthrough in the solutionof a problem， e.g. the Berry-Esseen inequality opens a way to evaluatethe 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 frequentlyused inequalities， such as the Jensen inequality， Schwarz inequality， Fa-tou Lemma， etc. Another annoyance for beginners is that an inequalitymay have many different names and reference sources. For example，the Schwarz inequality is also called the Cauchy， Cauchy-Schwarz orMinkovski-Bnyakovski inequality. Bennet， Hoeffding and Bernstein in-equalities have a very close relationship and format， and in literaturesome authors cross-cite in their use of the inequalities. This may be dueto one author using an inequality and subsequent authors just simplycopying the inequalitys format and its reference without checking theoriginal reference. All this may distress beginners very much. The aim of this book is to help beginners with these problems. Weprovide a place to find the most frequently used inequalities， their proofs（if not too lengthy） and some references. Of course， for some of the morepopularly known inequalities， such as Jensen and Schwarz， there is nonecessity to give a reference and we will not do so.
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.
Prof. Zhengyan Lin is a fellow of the Institute of Mathematical Statistics and currently a professor at Zhejiang University， Hangzhou， China. He is the prize winner of National Natural Science Award of China in 1997. Prof. Zhidong Bai is a fellow of TWAS and the Institute of Mathematical Statistics; he is a professor at the National University of Singapore and Northeast Normal University， Changchun， China.
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) ReferencesChapter 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 ReferencesChapter 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 ReferencesChapter 4 Estimates of the Difference of Two Distribution Functions 4.1 Fourier Transformation 4.2 Stein-Chen Method 4.3 Stieltjes Transformation ReferencesChapter 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 ReferencesChapter 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 ReferencesChapter 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 ReferencesChapter 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 ReferencesChapter 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 ReferencesChapter 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 ReferencesChapter 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 ReferencesChapter 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
