Almost Sure Behaviour of Partial Maxima Sequences of Some m-Dependent Stationary Sequences

2001 ◽  
pp. 243-249
Author(s):  
George Haiman ◽  
Lhassan Habach
Keyword(s):  
Stationary Sequences

ON THE SPECTRAL MEASURES OF SOME WEAKLY STATIONARY SEQUENCES INVOLVING RANDOMLY SPACED OBSERVATIONS

2012 ◽  
Vol 12 (01) ◽  
pp. 1150004
Author(s):  
RICHARD C. BRADLEY
Keyword(s):  
Spectral Density ◽  
Density Function ◽  
Spectral Density Function ◽  
Strong Mixing ◽  
Spectral Measures ◽  
Mixing Conditions ◽  
Stationary Sequences ◽  
Second Moments ◽  
The Central Limit Theorem ◽  
Complex Valued

In an earlier paper by the author, as part of a construction of a counterexample to the central limit theorem under certain strong mixing conditions, a formula is given that shows, for strictly stationary sequences with mean zero and finite second moments and a continuous spectral density function, how that spectral density function changes if the observations in that strictly stationary sequence are "randomly spread out" in a particular way, with independent "nonnegative geometric" numbers of zeros inserted in between. In this paper, that formula will be generalized to the class of weakly stationary, mean zero, complex-valued random sequences, with arbitrary spectral measure.

scholarly journals Renewal theorem for a class of stationary sequences

1986 ◽  
Vol 72 (2) ◽  
pp. 195-213 ◽  
Author(s):  
S. P. Lalley
Keyword(s):  
Stationary Sequences ◽  
Renewal Theorem

Stationary Sequences

2013 ◽  
pp. 493-506
Author(s):  
Alexandr A. Borovkov
Keyword(s):  
Stationary Sequences

Examples of Stationary Sequences with Approximate Negative Dependence

2019 ◽  
pp. 305-318
Author(s):  
Florence Merlevède ◽  
Magda Peligrad ◽  
Sergey Utev
Keyword(s):  
Spectral Density ◽  
Point Process ◽  
Point Processes ◽  
Random Variables ◽  
Stationary Processes ◽  
Negative Dependence ◽  
Stationary Sequences ◽  
Determinantal Point Process ◽  
Dependent Random Variables ◽  
Associated Sequences

In this chapter, we treat several examples of stationary processes which are asymptotically negatively dependent and for which the results of Chapter 9 apply. Many systems in nature are complex, consisting of the contributions of several independent components. Our first examples are functions of two independent sequences, one negatively dependent and one interlaced mixing. For instance, the class of asymptotic negatively dependent random variables is used to treat functions of a determinantal point process and a Gaussian process with a positive continuous spectral density. Another example is point processes based on asymptotically negatively or positively associated sequences and displaced according to a Gaussian sequence with a positive continuous spectral density. Other examples include exchangeable processes, the weighted empirical process, and the exchangeable determinantal point process.

Introduction to Stochastic Processes

2019 ◽  
pp. 1-24
Author(s):  
Florence Merlevède ◽  
Magda Peligrad ◽  
Sergey Utev
Keyword(s):  
Dynamical Systems ◽  
Stochastic Processes ◽  
Gaussian Approximation ◽  
Partial Sums ◽  
Stationary Sequences ◽  
Linear Processes ◽  
Limiting Behavior ◽  
Additive Functionals ◽  
Moderate Deviations Principle ◽  
Recursive Sequences

We start by stating the need for a Gaussian approximation for dependent structures in the form of the central limit theorem (CLT) or of the functional CLT. To justify the need to quantify the dependence, we introduce illustrative examples: linear processes, functions of stationary sequences, recursive sequences, dynamical systems, additive functionals of Markov chains, and self-interactions. The limiting behavior of the associated partial sums can be handled with tools developed throughout the book. We also present basic notions for stationary sequences of random variables: various definitions and constructions, and definitions of ergodicity, projective decomposition, and spectral density. Special attention is given to dynamical systems, as many of our results also apply in this context. The chapter also surveys the basic theory of the convergence of stochastic processes in distribution, and introduces the reader to tightness, finite-dimensional convergence, and the need for maximal inequalities. It ends with the concepts of the moderate deviations principle and its functional form.

Limit laws for the maxima of a class of quasi-stationary sequences

1983 ◽  
Vol 20 (04) ◽  
pp. 814-821 ◽  
Author(s):  
K. F. Turkman ◽  
A. M. Walker
Keyword(s):  
Random Variables ◽  
Extreme Value ◽  
Extreme Value Distributions ◽  
Stationary Sequences ◽  
Limiting Distributions ◽  
Limit Laws

A class of quasi-stationary sequences of random variables is introduced. After giving the definition, it is shown that the limiting distributions of the maxima of such sequences, when suitably normalized, converge to one of the three extreme-value distributions.

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