Probability Inequalities

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.

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.

Elementary Inequalities of Probabilities of Events	p. 1
Inclusion-exclusion Formula	p. 1
Corollaries of the Inclusion-exclusion Formula	p. 2
Further Consequences of the Inclusion-exclusion Formula	p. 2
Inequalities Related to Symmetric Difference	p. 6
Inequalities Related to Independent Events	p. 6
Lower Bound for Union	p. 8
References	p. 8
Inequalities Related to Commonly Used Distributions	p. 9
Inequalities Related to the Normal d.f.	p. 9
Slepian Type Inequalities	p. 12
Anderson Type Inequalities	p. 17
Khatri-¿idák Type Inequalities	p. 18
Corner Probability of Normal Vector	p. 19
Normal Approximations of Binomial and Poisson Distributions	p. 20
References	p. 22
Inequalities Related to Characteristic Functions	p. 23
Inequalities Related Only with c.f.	p. 23
Inequalities Related to c.f. and d.f.	p. 26
Normality Approximations of c.f. of Independent Sums	p. 27
References	p. 28
Estimates of the Difference of Two Distribution Functions	p. 29
Fourier Transformation	p. 29
Stein-Chen Method	p. 33
Stieltjes Transformation	p. 34
References	p. 36
Probability Inequalities of Random Variables	p. 37
Inequalities Related to Two r.v.'s	p. 37
Perturbation Inequality	p. 39
Symmetrization Inequalities	p. 40
Lévy Inequality	p. 41
Bickel Inequality	p. 42
Upper Bounds of Tail Probabilities of Partial Sums	p. 44
Lower Bounds of Tail Probabilities of Partial Sums	p. 44
Tail Probabilities for Maximum Partial Sums	p. 45
Tail Probabilities for Maximum Partial Sums (Continuation)	p. 46
Reflection Inequality of Tail Probability	p. 47
Probability of Maximal Increment	p. 48
Mogulskii Minimal Inequality	p. 49
Inequality	p. 50
References	p. 50
Bounds of Probabilities in Terms of Moments	p. 51
Chebyshev-Markov Type Inequalities	p. 51
Lower Bounds	p. 52
Series of Tail Probabilities	p. 53
Kolmogorov Type Inequalities	p. 54
Generalization of Kolmogorov Inequality for a Submartingale	p. 56
Rényi-Hájek Type Inequalities	p. 57
Chernoff Inequality	p. 60
Nagaev FukInequality	p. 62
Burkholder Inequality	p. 63
Complete Convergence of Partial Sums	p. 65
References	p. 65
Exponential Type Estimates of Probabilities	p. 67
Equivalence of Exponential Estimates	p. 67
Petrov Exponential Inequalities	p. 68
Hoeffding Inequality	p. 70
Bennett Inequality	p. 72
Bernstein Inequality	p. 73
Exponential Bounds for Sums of Bounded Variables	p. 74
Kolmogorov Inequalities	p. 75
Prokhorov Inequality	p. 79
Exponential Inequalities by Censoring	p. 80
Tail Probability of Weighted Sums	p. 82
References	p. 83
Moment Inequalities Related to One or Two Variables	p. 84
Moments of Truncation	p. 84
Exponential Moment of Bounded Variables	p. 84
Hölder Type Inequalities	p. 85
Jensen Type Inequalities	p. 86
Dispersion Inequality of Censored Variables	p. 87
Monotonicity of Moments of Sums	p. 88
Symmetrization Moment Inequatilies	p. 88
Inequality	p. 89
Exponential Moment of Normal Variable	p. 90
Inequatilies of Nonnegative Variable	p. 93
Inequality	p. 94
Exponential Moment of Upper Truncated Variables	p. 95
References	p. 96
Moment Estimates of (Maximum of) Sums of Random Variables	p. 97
Elementary Inequalities	p. 97
Type Inequalities	p. 99
The Case 1 ≤ r ≤ 2	p. 100
The Case r ≥ 2	p. 101
Jack-knifed Variance	p. 105
Khintchine Inequality	p. 106
Marcinkiewicz-Zygmund-Burkholder Type Inequalities	p. 108
Skorokhod Inequalities	p. 111
Moments of Weighted Sums	p. 112
Doob Crossing Inequalities	p. 113
Moments of Maximal Partial Sums	p. 114
Doob Inequalities	p. 115
Equivalence Conditions for Moments	p. 117
Inequalities	p. 123
Average Fill Rate	p. 127
References	p. 128
Inequalities Related to Mixing Sequences	p. 130
Covariance Estimates for Mixing Sequences	p. 131
Tail Probability on ¿-mixing Sequence	p. 134
Estimates of 4-th Moment on ¿-mixing Sequence	p. 135
Estimates of Variances of Increments of ¿-mixing Sequence	p. 136
Bounds of 2 + ¿-th Moments of Increments of ¿-mixing Sequence	p. 138
Tail Probability on ¿-mixing Sequence	p. 140
Bounds of 2 + ¿-th Moment of Increments of ¿-mixing Sequence	p. 142
Exponential Estimates of Probability on ¿-mixing Sequence	p. 143
References	p. 148
Inequalities Related to Associative Variables	p. 149
Covariance of PQD Varaibles	p. 149
Probability of Quadrant on PA (NA) Sequence	p. 150
Estimates of c.f.'s on LPQD (LNQD) Sequence	p. 151
Maximal Partial Sums of PA Sequence	p. 152
Variance of Increment of LPQD Sequence	p. 153
Expectation of Convex Function of Sum of NA Sequence	p. 154
Inequality for NA Sequence	p. 156
References	p. 157
Inequalities about Stochastic Processes and Banach Space Valued Random Variables	p. 158
Probability Estimates of Supremums of a Wiener Process	p. 158
Probability Estimate of Supremum of a Poisson Process	p. 162
Inequality	p. 164
Inequality	p. 167
Tail Probability of Gaussian Process	p. 168
Tail Probability of Randomly Signed Independent Processes	p. 168
Tail Probability of Adaptive Process	p. 170
Tail Probability on Submartingale	p. 172
Tail Probability of Independent Sum in B-Space	p. 173
Isoperimetric Inequalities	p. 173
Inequality	p. 174
Tail Probability of Normal Variable in B-Space	p. 175
Gaussian Measure on Symmetric Convex Sets	p. 175
Equivalence of Moments of B-Gaussian Variables	p. 176
Contraction Principle	p. 176
Symmetrization Inequalities in B-Space	p. 178
Decoupling Inequality	p. 178
References	p. 180
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