Abstract

For a L\'evy process $X_t$ with quadratic variation process $V_t=\sigma^2 t+ \sum_{00$, 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$X_{t}/\sqrt{V_t}$. As a main application it is shown that$X_{t}/\sqrt{V_t}\Rightarrow N(0,1)$, a standard normal random variable, as$t\downarrow 0$, if and only if$X_t/b(t)\Rightarrow N(0,1)$, as$t\downarrow 0$, for some non-stochastic 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 L\'evy measure of$X$.



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