A study of the Antlion Random Walk

Random walks (RWs) are fundamental stochastic processes with applications across physics, computer science, and information processing. A recent extension, the laser chaos decision-maker, employs chaotic time series from semiconductor lasers to solve multi-armed bandit (MAB) problems at ultrafast speeds, and its threshold adjustment mechanism has been modeled as an RW. However, previous analyses assumed complete memory preservation ($\alpha = 1$), overlooking the role of partial memory in balancing exploration and exploitation. In this paper, we introduce the Antlion Random Walk (ARW), defined by $X_t = \alpha X_{t-1} + \xi_t$ with $\alpha \in [0,1]$ and Rademacher-distributed increments $(\xi_t)$, which describes a walker pulled back toward the origin before each step. We show that varying $\alpha$ significantly alters ARW dynamics, yielding distributions that range from uniform-like to normal-like. Through mathematical and numerical analyses, we investigate expectation, variance, reachability, positive-side residence time, and distributional similarity. Our results place ARWs within the framework of autoregressive (AR(1)) processes while highlighting distinct non-Gaussian features, thereby offering new theoretical insights into memory-aware stochastic modeling of decision-making systems.