Dependence in Probability and Statistics

Patrice Bertail, Stéphan Clémençon (auth.), Patrice Bertail, Philippe Soulier, Paul Doukhan (eds.)9780387317410, 0387317414

This book gives a detailed account of some recent developments in the field of probability and statistics for dependent data. The book covers a wide range of topics from Markov chain theory and weak dependence with an emphasis on some recent developments on dynamical systems, to strong dependence in times series and random fields. A special section is devoted to statistical estimation problems and specific applications. The book is written as a succession of papers by some specialists of the field, alternating general surveys, mostly at a level accessible to graduate students in probability and statistics, and more general research papers mainly suitable to researchers in the field.

The first part of the book considers some recent developments on weak dependent time series, including some new results for Markov chains as well as some developments on new notions of weak dependence. This part also intends to fill a gap between the probability and statistical literature and the dynamical system literature. The second part presents some new results on strong dependence with a special emphasis on non-linear processes and random fields currently encountered in applications. Finally, in the last part, some general estimation problems are investigated, ranging from rate of convergence of maximum likelihood estimators to efficient estimation in parametric or non-parametric time series models, with an emphasis on applications with non-stationary data.

Patrice Bertail is researcher in statistics at CREST-ENSAE, Malakoff and Professor of Statistics at the University-Paris X. Paul Doukhan is researcher in statistics at CREST-ENSAE, Malakoff and Professor of Statistics at the University of Cergy-Pontoise. Philippe Soulier is Professor of Statistics at the University-Paris X.

Table of contents :
Front Matter….Pages 1-1
Regeneration-based statistics for Harris recurrent Markov chains….Pages 3-54
Subgeometric ergodicity of Markov chains….Pages 55-64
Limit Theorems for Dependent U-statistics….Pages 65-86
Recent results on weak dependence for causal sequences. Statistical applications to dynamical systems…..Pages 87-104
Parametrized Kantorovich-Rubinštein theorem and application to the coupling of random variables….Pages 105-121
Exponential inequalities and estimation of conditional probabilities….Pages 123-140
Martingale approximation of non adapted stochastic processes with nonlinear growth of variance….Pages 141-156
Front Matter….Pages 157-157
Almost periodically correlated processes with long memory….Pages 159-194
Long memory random fields….Pages 195-220
Long Memory in Nonlinear Processes….Pages 221-244
A LARCH(∞) Vector Valued Process….Pages 245-258
On a Szegö type limit theorem and the asymptotic theory of random sums, integrals and quadratic forms….Pages 259-286
Aggregation of Doubly Stochastic Interactive Gaussian Processes and Toeplitz forms of U -Statistics….Pages 287-302
Front Matter….Pages 303-303
On Efficient Inference in GARCH Processes….Pages 305-327
Almost sure rate of convergence of maximum likelihood estimators for multidimensional diffusions….Pages 329-347
Convergence rates for density estimators of weakly dependent time series….Pages 349-372
Variograms for spatial max-stable random fields….Pages 373-390
A non-stationary paradigm for the dynamics of multivariate financial returns….Pages 391-429
Multivariate Non-Linear Regression with Applications….Pages 431-473
Nonparametric estimator of a quantile function for the probability of event with repeated data….Pages 475-489