It was recently proved that the exponential decreasing rate of the probability that a random walk stays in a $d$-dimensional orthant is given by the minimum on this orthant of the Laplace transform of the random walk increments, provided that this minimum exists. In other cases, the random walk is “badly oriented” and the exponential rate may depend on the starting point $x$. We show here that this rate is nevertheless asymptotically equal to the infimum of the Laplace transform, as some selected coordinates of $x$ tend to infinity.
"On the exit time from an orthant for badly oriented random walks." Braz. J. Probab. Stat. 32 (1) 117 - 146, February 2018. https://doi.org/10.1214/16-BJPS334