A Functional Limit Theorem for Random Walk Conditioned to Stay Non-negative

Bryn-Jones, A. and Doney, R. A. (2006) A Functional Limit Theorem for Random Walk Conditioned to Stay Non-negative. [MIMS Preprint]

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Abstract

In this paper we consider an aperiodic integer-valued random walk S and a process S* which is an harmonic transform of S killed when it first enters the negative half; informally S* is "S conditioned to stay non-negative". If S is in the domain of attraction of the standard Normal law, without centring, a suitably normed and linearly interpolated version of S converges weakly to standard Brownian motion, and our main result is that under the same assumptions a corresponding statement holds for S*; the limit of course being the 3-dimensional Bessel process. Since this process can be thought of as Brownian motion conditioned to stay non-negative, in essence we our result shows that the interchange of the two limit operations is valid. We also establish some related results, including a local limit theorem for S*; and a bivariate renewal theorem for the ladder time and height process, which may be of independent interest.

Item Type:	MIMS Preprint
Subjects:	MSC 2010, the AMS's Mathematics Subject Classification > 60 Probability theory and stochastic processes
Depositing User:	Dr Peter Neal
Date Deposited:	05 Apr 2006
Last Modified:	08 Nov 2017 18:18
URI:	https://eprints.maths.manchester.ac.uk/id/eprint/207

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