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Hitting time of a half-line by two-dimensional random walk

Research paper by Yasunari Fukai

Indexed on: 02 Jan '04Published on: 02 Jan '04Published in: Probability Theory and Related Fields



Abstract

We consider the probability that a two-dimensional random walk starting from the origin never returns to the half-line {(x1,x2)|x1≤0,x2=0} before time n. It is proved that for aperiodic random walk with mean zero and finite 2+δ(>2)-th absolute moment, this probability times n1/4 converges to some positive constant c* as \({{n \rightarrow \infty}}\). We show that c* is expressed by using the characteristic function of the increment of the random walk. For the simple random walk, this expression gives \({{ c^{{*}} = \sqrt{{1+ \sqrt{{2}}}}/(2 \Gamma (3/4)).}}\)