Small-time compactness and convergence behavior of deterministically and self-normalised Lévy processes

Consider a Lévy process $X_t$ with quadratic variation process $V_t=\sigma^2 t+ \sum_{0<s\le t} (\Delta X_s)^2$, $t>0$, where $\Delta X_t=X_t-X_{t-}$ denotes the jump process of $X$. We give stability and compactness results, as $t \downarrow 0$, for the convergence both of the deterministically normed (and possibly centered) processes $X_t$ and $V_t$, as well as theorems concerning the ``self-normalised'' process $X_{t}/\sqrt{V_t}$. Thus, we consider the stochastic compactness and convergence in distribution of the 2-vector $\left((X_t-a(t))/b(t), V_t/b(t)\right)$, for deterministic functions $a(t)$ and $b(t)>0$, as $t \downarrow 0$, possibly through a subsequence; and the stochastic compactness and convergence in distribution of $X_{t}/\sqrt{V_t}$, possibly to a nonzero constant (for stability), as $t \downarrow 0$, again possibly through a subsequence.

As a main application it is shown that $X_{t}/\sqrt{V_t}\stackrel{\mathrm{D}}{\longrightarrow} N(0,1)$, a standard normal random variable, as $t \downarrow 0$, if and only if $X_t/b(t)\stackrel{\mathrm{D}}{\longrightarrow} N(0,1)$, as $t\downarrow0$, for some nonstochastic function $b(t)>0$; thus, $X_t$ is in the domain of attraction of the normal distribution, as $t \downarrow 0$, with or without centering constants being necessary (these being equivalent).

We cite simple analytic equivalences for the above properties, in terms of the Lévy measure of $X$. Functional versions of the convergences are also given.