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

Abstract

Consider a Levy process Xt with quadratic variation process Vt = σ2t+Σ0<s≤t(ΔXs)2, t> 0, where ΔXt = Xt -Xt- 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 Xt and Vt, as well as theorems concerning the "self-normalised" process Xt//√V t- Thus, we consider the stochastic compactness and convergence in distribution of the 2-vector ((Xt - a(t))/b(t),Vt/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 Xt√/Vt, possibly to a nonzero constant (for stability), as t ↕ 0, again possibly through a subsequence. As a main application it is shown that Xt//√V→t D-N(0,1), a standard normal random variable, as t ↕ 0, if and only if Xt/b(t) →t N(0, 1), as t ↕ 0, for some nonstochastic function b(t) > 0; thus, Xt 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. Show more