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Consider a nearest-neighbor random walk with certain asymptotically zerodrift on the positive half line. Let $M$ be the maximum of an excursionstarting from $1$ and ending at $0.$ We study the distribution of $M$ andcharacterize its asymptotics, which is quite different from those of simplerandom walks.

On a maximum of nearest-neighbor random walk with asymptotically zero drift on lattice of positive half line