Limit Theorems for Weighted Sums of Infinite Variance Random Variables Attracted to Integrals of Linear Fractional Stable Motions

Abstract

Let $\{\xi_j\}_{j\in\mathbb{Z}}$ be a sequence of random variables which belong to the domain of attraction of a linear fractional stable motion $\{\Delta_{H,\alpha}(t)\}$ with infinite variance. We study the convergence of weighted sums $I_n(f):=A_n\sum_{j\in\mathbb{Z}}f({j}/{n})\xi_j$ with a suitable scaling $A_n$, to $I(f):=\int_{-\infty}^{\infty}f(u)d\Delta_{H,\alpha}(u)$ in distribution under suitable assumptions on a class of deterministic functions $f$. We also show that if $\{f_t, t\ge 0\}$ are the kernel functions from the ``moving average'' representation of a linear fractional stable motion with another index $H'$, then $\{I_n(f_t)\}$ converges to a linear fractional stable motion $\{ \Delta _{H+H'-1/\alpha, \alpha}(t)\}$.