Abstract.
We consider the probability that a two-dimensional random walk starting from the origin never returns to the half-line {(x 1 ,x 2 )|x 1 ≤0,x 2 =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 n 1/4 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 \({{ c^{{*}} = \sqrt{{1+ \sqrt{{2}}}}/(2 \Gamma (3/4)).}}\)
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Mathematics Subject Classification (2000): 60G50, 60E10
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Fukai, Y. Hitting time of a half-line by two-dimensional random walk. Probab. Theory Relat. Fields 128, 323–346 (2004). https://doi.org/10.1007/s00440-003-0306-y
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DOI: https://doi.org/10.1007/s00440-003-0306-y