ACTA issues

Convergence in distribution of Lévy processes at small times with self-normalization

Ross Maller, David M. Mason

Acta Sci. Math. (Szeged) 74:1-2(2008), 315-347

Abstract. Let $X_t$ be a Lévy process and $V_t=\sigma ^2 t+ \sum_{0< s\le t} (\Delta X_s)^2, t>0$, its quadratic variation process, where $\Delta X_t=X_t-X_{t-} $ denotes the jump process of $X$. When $X$ is symmetric, we show that the self-normalized process $Y_t:=X_{t}/\sqrt{V_t}$ converges in distribution as $t\downarrow0$ to an a.s. finite, nondegenerate random variable, if and only if (i) $X_t$ is in the domain of attraction of a nondegenerate stable random variable $S_\alpha $; that is, if and only if, for some nonstochastic function $b(t)>0$, $X_t/b(t)\mathop{\longrightarrow}^{\mathrm D} S_\alpha $, as $t\downarrow0$; or else, (ii) the tail of the Lévy measure of $X$ is slowly varying at 0. This is proved as an application of criteria we set out for the joint convergence of $X_t$ and $V_t$, after norming (and centering, in the case of $X$), to infinitely divisible, and, in particular, to stable, limit rvs, as $t\downarrow0$, either continuously, or through a subsequence.

AMS Subject Classification (1991): 60F05, 60F15, 60G51, 62E20, 62G30

Keyword(s): Lévy process, self-normalized, small-time behavior, domain of attraction

Received July 4, 2007, and in revised form February 11, 2008. (Registered under 6019/2009.)