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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.
目錄
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

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