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In this paper, we consider the $(1,R)$ state-dependent reflecting random walk(RW) on the half line, allowing the size of jumps to the right at maximal $R$and to the left only 1. We provide an explicit criterion for positiverecurrence and the explicit expression of the stationary distribution based onthe intrinsic branching structure within the walk. As an application, we obtainthe tail asymptotic of the stationary distribution in the "near critical"situation.

Tail asymptotic of the stationary distribution for the state dependent (1,R)-reflecting random walk: near critical