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Linear fractional stable motion: A wavelet estimator of the α parameter

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  • Ayache, Antoine
  • Hamonier, Julien

Abstract

Linear fractional stable motion, denoted by {XH,α(t)}t∈R, is one of the most classical stable processes; it depends on two parameters H∈(0,1) and α∈(0,2). The parameter H characterizes the self-similarity property of {XH,α(t)}t∈R while the parameter α governs the tail heaviness of its finite dimensional distributions; throughout our article we assume that the latter distributions are symmetric, that H>1/α and that H is known. We show that, on the interval [0,1], the asymptotic behavior of the maximum, at a given scale j, of absolute values of the wavelet coefficients of {XH,α(t)}t∈R, is of the same order as 2−j(H−1/α); then we derive from this result a strongly consistent (i.e. almost surely convergent) statistical estimator for the parameter α.

Suggested Citation

  • Ayache, Antoine & Hamonier, Julien, 2012. "Linear fractional stable motion: A wavelet estimator of the α parameter," Statistics & Probability Letters, Elsevier, vol. 82(8), pages 1569-1575.
  • Handle: RePEc:eee:stapro:v:82:y:2012:i:8:p:1569-1575
    DOI: 10.1016/j.spl.2012.04.005
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    References listed on IDEAS

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    1. Delbeke, Lieve & Abry, Patrice, 2000. "Stochastic integral representation and properties of the wavelet coefficients of linear fractional stable motion," Stochastic Processes and their Applications, Elsevier, vol. 86(2), pages 177-182, April.
    2. Stilian Stoev & Murad S. Taqqu, 2005. "Asymptotic self-similarity and wavelet estimation for long-range dependent fractional autoregressive integrated moving average time series with stable innovations," Journal of Time Series Analysis, Wiley Blackwell, vol. 26(2), pages 211-249, March.
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    Cited by:

    1. Mazur, Stepan & Otryakhin, Dmitry & Podolskij, Mark, 2018. "Estimation of the linear fractional stable motion," Working Papers 2018:3, Örebro University, School of Business.

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