Abstract
In this paper we estimate both the Hurst and the stability indices of a $H$-self-similar stable process. More precisely, let $X$ be a $H$-sssi (self-similar stationary increments) symmetric $\alpha$-stable process. The process $X$ is observed at points $\frac{k}{n}$, $k=0,\ldots,n$. Our estimate is based on $\beta$-negative power variations with $-\frac{1}{2}<\beta<0$. We obtain consistent estimators, with rate of convergence, for several classical $H$-sssi $\alpha$-stable processes (fractional Brownian motion, well-balanced linear fractional stable motion, Takenaka’s process, Levy motion). Moreover, we obtain asymptotic normality of our estimators for fractional Brownian motion and Levy motion.
Highlights
Self-similar processes play an important role in probability because of their connection to limit theorems and they are widely used to model natural phenomena
Self-similar α-stable processes have been proposed to model some natural phenomena with heavy tails, as in [21] and references therein
The estimation of various indices of H−sssi α−stable processes has been a problem studied since several decades ago and, even nowadays, it continues to be a challenge
Summary
Self-similar processes play an important role in probability because of their connection to limit theorems and they are widely used to model natural phenomena. In the case of fractional Brownian motion, the estimation of the self-similarity index H has attracted attention to many authors and many methods have been proposed for solving this problem. For linear fractional stable motions, strongly consistent estimators of the self-similarity index H, based on the discrete wavelet transform of the processes, have been proposed without requirement that α to be known, as in [2], [20], [23], [24]. For linear multifractional stable motions, in [4], the authors presented strongly consistent estimators of the localisability function H(.) and the stability index α using wavelet coefficients when α ∈ (1, 2) and H(.) is a Holder function smooth enough, with values in a compact subinterval [H, H] of (1/α, 1).
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