Premium
A Functional Limit Theorem for Random Walk Conditioned to Stay Non‐Negative
Author(s) -
Bryn-Jones A.,
Doney R. A.
Publication year - 2006
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610706022964
Subject(s) - mathematics , limit (mathematics) , brownian motion , random walk , bessel process , bivariate analysis , integer (computer science) , domain (mathematical analysis) , aperiodic graph , mathematical analysis , combinatorics , statistics , classical orthogonal polynomials , gegenbauer polynomials , computer science , orthogonal polynomials , programming language
In this paper we consider an aperiodic integer‐valued random walk S and a process S* that is a 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 three‐dimensional Bessel process. As this process can be thought of as Brownian motion conditioned to stay non‐negative, in essence 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.