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Random Walk Conditioned to Stay Positive
Author(s) -
Biggins J. D.
Publication year - 2003
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/s0024610702003708
Subject(s) - random walk , mathematics , markov chain , bessel process , heterogeneous random walk in one dimension , brownian motion , infinity , combinatorics , discrete mathematics , zero (linguistics) , continuous time random walk , statistical physics , statistics , mathematical analysis , physics , orthogonal polynomials , classical orthogonal polynomials , linguistics , gegenbauer polynomials , philosophy
A random walk that is certain to visit (0, ∞) has associated with it, via a suitable h ‐transform, a Markov chain called ‘random walk conditioned to stay positive’, which is defined properly below. In continuous time, if the random walk is replaced by Brownian motion then the analogous associated process is Bessel‐3. Let φ( x ) = log log x . The main result obtained in this paper, which is stated formally in Theorem 1, is that, when the random walk has zero mean and finite variance, the total time for which the random walk conditioned to stay positive is below x ultimately lies between Lx 2 /φ( x ) and Ux 2 φ( x ), for suitable (non‐random) positive L and finite U , as x goes to infinity. For Bessel‐3, the best L and U are identified.