Limit theorems for associated random fields and related systems

Alexander Bulinski, Alexey Shashkin9812709401, 9789812709400, 9789812709417

This volume is devoted to the study of asymptotic properties of wide classes of stochastic systems arising in mathematical statistics, percolation theory, statistical physics and reliability theory. Attention is paid not only to positive and negative associations introduced in the pioneering papers by Harris, Lehmann, Esary, Proschan, Walkup, Fortuin, Kasteleyn and Ginibre, but also to new and more general dependence conditions. Naturally, this scope comprises families of independent real-valued random variables. A variety of important results and examples of Markov processes, random measures, stable distributions, Ising ferromagnets, interacting particle systems, stochastic differential equations, random graphs and other models are provided. For such random systems, it is worthwhile to establish principal limit theorems of the modern probability theory (central limit theorem for random fields, weak and strong invariance principles, functional law of the iterated logarithm etc.) and discuss their applications. There are 434 items in the bibliography. The book is self-contained, provides detailed proofs, for reader s convenience some auxiliary results are included in the Appendix (e.g. the classical Hoeffding lemma, basic electric current theory etc.).

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Table of contents :
Contents……Page 10
Preface……Page 8
1 Basic definitions and simple examples……Page 12
2 Classes of associated and related systems……Page 28
3 Random measures……Page 48
4 Association and probability measures on lattices……Page 75
5 Further extensions of dependence notions……Page 99
2. Moment and Maximal Inequalities……Page 118
1 Bounds for partial sums in the Lp space……Page 119
2 Results based on supermodular order……Page 145
3 Rosenthal-type inequalities……Page 154
4 Estimates for the distribution functions of partial maxima……Page 171
3. Central Limit Theorem……Page 180
1 Sufficient conditions for normal approximation……Page 181
2 The Newman conjecture……Page 208
3 Sharp rates of normal approximation……Page 226
1 Strong law of large numbers……Page 240
2 Rate of convergence in the LLN……Page 244
3 Almost sure Gaussian approximation……Page 257
1 Weak invariance principle……Page 262
2 Strong invariance principle……Page 274
1 Extensions of the classical LIL……Page 294
2 Functional LIL……Page 309
3 Law of a single logarithm……Page 321
1 Statistics involving random normalization……Page 330
2 Kernel density estimation……Page 349
3 Empirical processes……Page 359
1 Stationary associated measures……Page 364
2 PDE with random initial data……Page 377
3 Asymptotical behavior of transformed solutions of the Burgers equation……Page 385
A.1 Extensions of the Hoe ding lemma……Page 394
A.2 Markov processes. Background……Page 396
A.3 Poisson spatial process……Page 400
A.4 Electric currents……Page 403
A.5 The Moricz theorem……Page 406
A.6 Gaussian approximation……Page 411
Bibliography……Page 422
Notation Index……Page 442
Subject Index……Page 444