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Inheritance of strong mixing and weak dependence under renewal sampling

Part of: Sampling theory, sample surveys Limit theorems Stochastic processes

Published online by Cambridge University Press:  14 October 2022

Dirk-Philip Brandes ,
Imma Valentina Curato Open the ORCID record for Imma Valentina Curato [Opens in a new window]  and
Robert Stelzer
Dirk-Philip Brandes*
Affiliation:
Ulm University
Imma Valentina Curato*
Affiliation:
Ulm University
Robert Stelzer*
Affiliation:
Ulm University
*
*Postal address: Helmholtzstraße 18, 89069 Ulm, Germany.
*Postal address: Helmholtzstraße 18, 89069 Ulm, Germany.
*Postal address: Helmholtzstraße 18, 89069 Ulm, Germany.
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Abstract

Let X be a continuous-time strongly mixing or weakly dependent process and let T be a renewal process independent of X. We show general conditions under which the sampled process $(X_{T_i},T_i-T_{i-1})^{\top}$ is strongly mixing or weakly dependent. Moreover, we explicitly compute the strong mixing or weak dependence coefficients of the renewal sampled process and show that exponential or power decay of the coefficients of X is preserved (at least asymptotically). Our results imply that essentially all central limit theorems available in the literature for strongly mixing or weakly dependent processes can be applied when renewal sampled observations of the process X are at our disposal.

Keywords

Renewal samplingstationary processesstationary random fields

MSC classification

Primary: 60G10: Stationary processes 62D05: Sampling theory, sample surveys
Secondary: 60F05: Central limit and other weak theorems 60G60: Random fields
Type
Original Article
Information
Journal of Applied Probability , Volume 60 , Issue 2 , June 2023 , pp. 435 - 451
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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