Asymptotic theory for quadratic variation of harmonizable fractional stable processes

Andreas Basse-O’connor; Mark Podolskij

doi:10.1090/tpms/1203

Asymptotic theory for quadratic variation of harmonizable fractional stable processes

Department of Mathematics

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Abstract

In this paper we study the asymptotic theory for quadratic variation of a harmonizable fractional α-stable process. We show a law of large numbers with a non-ergodic limit and obtain weak convergence towards a Lévy-driven Rosenblatt random variable when the Hurst parameter satisfies H ∈ (1/2, 1) and α(1−H) < 1/2. This result complements the asymptotic theory for fractional stable processes.

Original language	English
Journal	Theory of Probability and Mathematical Statistics
Volume	110
Pages (from-to)	3-12
Number of pages	10
ISSN	0094-9000
DOIs	https://doi.org/10.1090/tpms/1203
Publication status	Published - 2024

Keywords

Fractional processes
harmonizable processes
limit theorems
quadratic vari-ation
stable Lévy motion

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abstract = "In this paper we study the asymptotic theory for quadratic variation of a harmonizable fractional α-stable process. We show a law of large numbers with a non-ergodic limit and obtain weak convergence towards a L{\'e}vy-driven Rosenblatt random variable when the Hurst parameter satisfies H ∈ (1/2, 1) and α(1−H) < 1/2. This result complements the asymptotic theory for fractional stable processes.",

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AB - In this paper we study the asymptotic theory for quadratic variation of a harmonizable fractional α-stable process. We show a law of large numbers with a non-ergodic limit and obtain weak convergence towards a Lévy-driven Rosenblatt random variable when the Hurst parameter satisfies H ∈ (1/2, 1) and α(1−H) < 1/2. This result complements the asymptotic theory for fractional stable processes.

KW - Fractional processes

KW - harmonizable processes

KW - limit theorems

KW - quadratic vari-ation

KW - stable Lévy motion

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ER -

Asymptotic theory for quadratic variation of harmonizable fractional stable processes

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