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

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