## 2006.52: A Functional Limit Theorem for Random Walk Conditioned to Stay Non-negative

2006.52:
A. Bryn-Jones and R. A. Doney
(2006)
*A Functional Limit Theorem for Random Walk
Conditioned to Stay Non-negative.*

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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 |
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Subjects: | MSC 2000 > 60 Probability theory and stochastic processes |

MIMS number: | 2006.52 |

Deposited By: | Dr Peter Neal |

Deposited On: | 05 April 2006 |

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