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

Consider a Levy process Xt with quadratic variation process V_t = σ²t+Σ0<s≤t(ΔX_s)², t> 0, where ΔX_t = X_t -X_t- denotes the jump process of ×. We give stability and compactness results, as t ↕ 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//√V _t- Thus, we consider the stochastic compactness and convergence in distribution of the 2-vector ((Xt - a(t))/b(t),V_t/b(t)), for deterministic functions a(t)and b(t) > 0, as t ↕ 0, possibly through a subsequence; and the stochastic compactness and convergence in distribution of X_t√/V_t, possibly to a nonzero constant (for stability), as t ↕ 0, again possibly through a subsequence. As a main application it is shown that X_t//√V→t D-N(0,1), a standard normal random variable, as t ↕ 0, if and only if X_t/b(t) →t N(0, 1), as t ↕ 0, for some nonstochastic function b(t) > 0; thus, X_t is in the domain of attraction of the normal distribution, as t ↕ 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 ×. Functional versions of the convergences are also given.