scholarly journals Density convergence in the Breuer–Major theorem for Gaussian stationary sequences

Bernoulli ◽  
2015 ◽  
Vol 21 (4) ◽  
pp. 2336-2350 ◽  
Author(s):  
Yaozhong Hu ◽  
David Nualart ◽  
Samy Tindel ◽  
Fangjun Xu
Keyword(s):  
Stationary Sequences ◽  
Major Theorem

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 The Breuer–Major theorem in total variation: Improved rates under minimal regularity

2021 ◽  
Vol 131 ◽  
pp. 1-20
Author(s):  
Ivan Nourdin ◽  
David Nualart ◽  
Giovanni Peccati
Keyword(s):  
Total Variation ◽  
Major 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.

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