Linear Multifractional Stable Motion：$H(\cdot)$和$\al$参数的小波估计
Linear Multifractional Stable Motion: wavelet estimation of $H(\cdot)$ and $\al$ parameters
摘要：
Hurst参数$H$和稳定性参数$\al$的线性分数稳定运动（LFSM）是著名的高斯分数布朗运动（FBM）在重尾稳定分布\cite{SamTaq,EmMa}上的最经典的推广。为了克服其应用领域的一些限制，来自其增量的平稳性以及其自相似指数随时间的恒定性，Stoev和Taqqu在\cite{stoev2004stochastic}中引入了LFSM的扩展，称为线性多分数稳定运动（LMSM），其中Hurst参数变为依赖于时间变量$t$的函数$H(\cdot)$。与LFSM相似，LMSM的边缘分布的尾部重度由$\al$决定，在一定条件下，其自相似性由$H(\cdot)$决定，路径粗糙度与$H(\cdot)-1/\al$密切相关.也就是说，在\cite{stoev2004stochastic}中表明，$H(t_0)$是LMSM在时间$t_0\neq 0$上的自相似指数;此外，最近在\cite{hamonier2012lmsm}中建立了量$\min_{t\in I} H(t)-1/\al$和$H(t_0)-1/\al$，分别是LMSM在紧区间$I$上的均匀Hölder指数和在$t_0$上的局部Hölder指数。我们的文章的主要目标，是构建，使用LMSM的小波系数，强一致的（即几乎必然收敛）的统计估计$\min_{t\in I} H(t)$，$H(t_0)$，和$\al$ ;我们的估计结果，当$\al\in (1,2)$，和$H(\cdot)$是一个霍德尔函数足够光滑，在一个紧凑的子区间$[\underline{H},\bar{H}]$的$(1/\al,1)$的值。
英文摘要：
Linear Fractional Stable Motion (LFSM) of Hurst parameter $H$ and of stability parameter $\al$, is one of the most classical extensions of the well-known Gaussian Fractional Brownian Motion (FBM), to the setting of heavy-tailed stable distributions \cite{SamTaq,EmMa}. In order to overcome some limitations of its areas of application, coming from stationarity of its increments as well as constancy over time of its self-similarity exponent, Stoev and Taqqu introduced in \cite{stoev2004stochastic} an extension of LFSM, called Linear Multifractional Stable Motion (LMSM), in which the Hurst parameter becomes a function $H(\cdot)$ depending on the time variable $t$. Similarly to LFSM, the tail heaviness of the marginal distributions of LMSM is determined by $\al$; also, under some conditions, its self-similarity is governed by $H(\cdot)$ and its path roughness is closely related to $H(\cdot)-1/\al$. Namely, it was shown in \cite{stoev2004stochastic} that $H(t_0)$ is the self-similarity exponent of LMSM at a time $t_0\neq 0$; moreover, very recently, it was established in \cite{hamonier2012lmsm}, that the quantities $\min_{t\in I} H(t)-1/\al$, and $H(t_0)-1/\al$, are respectively the uniform Hölder exponent of LMSM on a compact interval $I$, and its local Hölder exponent at $t_0$. The main goal of our article, is to construct, using wavelet coefficients of LMSM, strongly consistent (i.e. almost surely convergent) statistical estimators of $\min_{t\in I} H(t)$, $H(t_0)$, and $\al$; our estimation results, are obtained when $\al\in (1,2)$, and, $H(\cdot)$ is a Hölder function smooth enough, with values in a compact subinterval $[\underline{H},\bar{H}]$ of $(1/\al,1)$.
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