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

DOI:
https://doi.org/10.1090/S0002-9947-09-05032-6

Published electronically:
November 18, 2009

MathSciNet review:
2574893

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Abstract | References | Similar Articles | Additional Information

Abstract: Consider a Lévy process with quadratic variation process , , where denotes the jump process of . We give stability and compactness results, as , for the convergence both of the deterministically normed (and possibly centered) processes and , as well as theorems concerning the ``self-normalised'' process . Thus, we consider the stochastic compactness and convergence in distribution of the 2-vector , for deterministic functions and , as , possibly through a subsequence; and the stochastic compactness and convergence in distribution of , possibly to a nonzero constant (for stability), as , again possibly through a subsequence.

As a main application it is shown that , a standard normal random variable, as , if and only if , as , for some nonstochastic function ; thus, is in the domain of attraction of the normal distribution, as , 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.

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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

Email:
Ross.Maller@anu.edu.au

**David M. Mason**

Affiliation:
Food and Resource Economics, University of Delaware, 206 Townsend Hall, Newark, Delaware 19717

Email:
davidm@Udel.Edu

DOI:
https://doi.org/10.1090/S0002-9947-09-05032-6

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