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Rates for the SLLN for long-memory and heavy-tailed processes

Part of: Stochastic processes Distribution theory - Probability Limit theorems

Published online by Cambridge University Press:  24 June 2025

Samir Ben Hariz  and
Salim Bouzebda Open the ORCID record for Salim Bouzebda [Opens in a new window]
Samir Ben Hariz*
Affiliation:
Université du Maine
Salim Bouzebda*
Affiliation:
Université de technologie de Compiègne
*
*Postal address: Laboratoire de Statistique et Processus, Département de Mathématiques, Université du Maine, Av. Olivier Messiaen BP 535 72017 Le Mans CEDEX, France. Email: Samir.Ben_Hariz@univ-lemans.fr
**Postal address: Université de technologie de Compiègne, LMAC (Laboratory of Applied Mathematics of Compiègne), 57 Avenue de Landshut CS 60319 60203 Compiègne CEDEX, France. Email: Salim.Bouzebda@utc.fr
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Abstract

The present paper develops a unified approach when dealing with short- or long-range dependent processes with finite or infinite variance. We are concerned with the convergence rate in the strong law of large numbers (SLLN). Our main result is a Marcinkiewicz–Zygmund law of large numbers for $S_{n}(f)= \sum_{i=1}^{n}f(X_{i})$, where $\{X_i\}_{i\geq 1}$ is a real stationary Gaussian sequence and $f(\!\cdot\!)$ is a measurable function. Key technical tools in the proofs are new maximal inequalities for partial sums, which may be useful in other problems. Our results are obtained by employing truncation alongside new maximal inequalities. The result can help to differentiate the effects of long memory and heavy tails on the convergence rate for limit theorems.

Keywords

Long-range dependenceheavy tailsrate of convergencestrong law of large numbersfunctional of Gaussian sequencesmaximal inequalities

MSC classification

Primary: 60E15: Inequalities; stochastic orderings
Secondary: 60F15: Strong theorems 60G15: Gaussian processes

Information

Type
Original Article
Information
Journal of Applied Probability , Volume 62 , Issue 4 , December 2025 , pp. 1513 - 1533
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Applied Probability Trust

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