We consider the class of symmetric <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\alpha$</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-stable moving average processes with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$1< \alpha < 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo><</mo> <mi>α</mi> <mo><</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. These processes are H-self-similar (<InlineEquation ID="IEq3"> <EquationSource Format="TEX">$0< H < 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>H</mi> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>) with stationary increments, indexed by <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathbb {R}^{d}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation>, and driven by a symmetric <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\alpha$</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-stable random measure <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$M_{\alpha }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>M</mi> <mi>α</mi> </msub> </math></EquationSource> </InlineEquation>. Our objective is to characterize these processes by estimating the Hurst parameter H, utilizing estimators based on p-variations and wavelet decomposition techniques. The idea is to exploit the self-similarity structure to calculate the p-variation along fixed directions. Two main results will be presented: the first establishes a law of large numbers type estimator for H, while the second provides a central limit theorem demonstrating convergence either to a Gaussian or to a stable distribution.

Limit theorems for self-similar symmetric stable moving average processes: a study with p-variations

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发表日期：

2026年5月20日

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Marie-Eliette Dury,
Nourddine Azzaoui,
Arnaud Guillin

摘要

We consider the class of symmetric $\alpha$ $α$ -stable moving average processes with $1< \alpha < 2$ $1 < α < 2$ . These processes are H-self-similar ( $0< H < 1$ $0 < H < 1$ ) with stationary increments, indexed by $\mathbb {R}^{d}$ $R^{d}$ , and driven by a symmetric $\alpha$ $α$ -stable random measure $M_{\alpha }$ $M_{α}$ . Our objective is to characterize these processes by estimating the Hurst parameter H, utilizing estimators based on p-variations and wavelet decomposition techniques. The idea is to exploit the self-similarity structure to calculate the p-variation along fixed directions. Two main results will be presented: the first establishes a law of large numbers type estimator for H, while the second provides a central limit theorem demonstrating convergence either to a Gaussian or to a stable distribution.