  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.
