Moran random walk with reset and short memory

We investigated the statistical properties of the Moran random walk (Yₙ)ₙ in one dimension, focusing on short memory. Specifically, employing generating function techniques, we determined the cumulative distribution function and the mean of the height Hₙ. Furthermore, we derived explicit expressions for the distribution, mean, and variance of Yₙ, along with its asymptotic distribution. Finally, we provided the distribution of the waiting time τₕ, which represents the number of steps required to reach a specified level h, as the conclusion of our study.