Angular asymptotics for multi-dimensional non-homogeneous random walks with asymptotically zero drift

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We study the first exit time τ from an arbitrary cone with apex at the origin by a non-homogeneous random walk (Markov chain) on ᵈ (d ≥ 2) with mean drift that is asymptotically zero. Specifically, if the mean drift at ∈ ᵈ is of magnitude O(⁻¹), we show that τ<∞ a.s. for any cone. On the other hand, for an appropriate drift field with mean drifts of magnitude ⁻β, β∈ (0,1), we prove that our random walk has a limiting (random) direction and so eventually remains in an arbitrarily narrow cone. The conditions imposed on the random walk are minimal: we assume only a uniform bound on 2nd moments for the increments and a form of weak isotropy. We give several illustrative examples, including a random walk in random environment model.