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Functional Gaussian Approximation for Dependent Structures$
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Florence Merlevède, Magda Peligrad, and Sergey Utev

Print publication date: 2019

Print ISBN-13: 9780198826941

Published to Oxford Scholarship Online: April 2019

DOI: 10.1093/oso/9780198826941.001.0001

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Stationary Sequences in a Random Time Scenery

Stationary Sequences in a Random Time Scenery

Chapter:
(p.319) 11 Stationary Sequences in a Random Time Scenery
Source:
Functional Gaussian Approximation for Dependent Structures
Author(s):

Florence Merlevède

Magda Peligrad

Sergey Utev

Publisher:
Oxford University Press
DOI:10.1093/oso/9780198826941.003.0011

In this chapter, we analyze the asymptotic behavior of the partial sums process associated with examples of stationary sequences in a random time scenery. The examples considered are stationary sequences sampled by shifted renewal Markov chains and random walks in a strictly stationary scenery. The asymptotic behavior of the partial sums process is essentially investigated with the help of the weak invariance principles stated in Chapter 4. More precisely, for the partial sums process associated with a stationary process sampled by a renewal Markov chain stated at zero, due to the non-stationarity of the underlying sequence, the functional CLT is obtained as an application of the functional CLT for non-stationary sequences developed in Section 4.4. In the case where we are sampling a strictly stationary random scenery by a random walk, stationarity is preserved, and the invariance principle is then derived by using the functional CLT under Maxwell–Woodroofe condition.

Keywords:   renewal Markov chain, random walk, random scenery, projective criteria

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