Exit times from cones for non-homogeneous random walk with asymptotically zero drift

We study the first exit time τ from an arbitrary cone with apex at the origin by a non-homogeneous random walk on Z2 with mean drift that is asymptotically zero. Assuming bounded jumps and a form of weak isotropy, we give conditions for τ to be almost surely finite, and for the existence and non-existence of moments E[τp], p > 0. Specifically, if the mean drift at x ∈ Z2 is of magnitude O(‖x‖−1), then τ < ∞ a.s. for any cone, and we give polynomial estimates on the tail of τ , while a mean drift of order ‖x‖−β, β ∈ (0, 1), can lead to τ = ∞ with positive probability for any cone. For mean drift o(‖x‖−1) (such as in the driftless case) we essentially recover, amongst other things, the random walk analogue of a theorem for Brownian motion due to Spitzer.