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

Authors: Ross Maller and David M. Mason
Journal: Trans. Amer. Math. Soc. 362 (2010), 2205-2248
MSC (2000): Primary 60F05, 60F17, 60G51
Published electronically: November 18, 2009
MathSciNet review: 2574893
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Abstract: 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.

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Additional Information

Ross Maller
Affiliation: Centre for Mathematical Analysis & School of Finance and Applied Statistics, Australian National University, PO Canberra, ACT, Australia

David M. Mason
Affiliation: Food and Resource Economics, University of Delaware, 206 Townsend Hall, Newark, Delaware 19717
Email: davidm@Udel.Edu

Received by editor(s): June 10, 2008
Received by editor(s) in revised form: March 3, 2009
Published electronically: November 18, 2009
Additional Notes: The first author’s research was partially supported by ARC Grant DP0664603
The second author’s research was partially supported by NSF Grant DMS–0503908.
Article copyright: © Copyright 2009 American Mathematical Society

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