Random walks with drift inside a pyramid: convergence rate for the survival probability

TL;DR

This paper analyzes the convergence rate of the survival probability for multidimensional random walks with drift inside pyramids, providing a way to compute the exponential rate of convergence using Laplace transforms.

Contribution

It introduces a method to quantify the convergence speed of survival probabilities in pyramid-shaped cones for random walks with drift, linking it to Laplace transforms.

Findings

Convergence rate can be explicitly computed using min-max of Laplace transform.

Survival probability converges exponentially when drift is inside the cone.

Results are illustrated with various examples.

Abstract

We consider multidimensional random walks in pyramids, which by definition are cones formed by finite intersections of half-spaces. The main object of interest is the survival probability $\mathbb{P}(\tau>n)$, $\tau$ denoting the first exit time from a fixed pyramid. When the drift belongs to the interior of the cone, the survival probability sequence converges to the non-exit probability $\mathbb{P}(\tau=\infty)$, which is positive. In this note, we quantify the speed of convergence, and prove that the exponential rate of convergence may be computed by means of a certain min-max of the Laplace transform of the random walk increments. We illustrate our results with various examples.