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In this paper, we consider the $(L,1)$ state-dependent reflecting random walk(RW) on the half line, which is a RW allowing jumps to the left at a maxialsize $L$. For this model, we provide an explicit criterion for (positive)recurrence and an explicit expression for the stationary distribution.As anapplication, we prove the geometric tail asymptotic behavior of the stationarydistribution under certain conditions. The main tool employed in the paper isthe intrinsic branching structure within the $(L,1)$-random walk.

Explicit stationary distribution of the $(L,1)$-reflecting random walk on the half line