  We study the first exit time $\tau$ from an arbitrary cone with apex at the
origin by a non-homogeneous random walk (Markov chain) on $\Z^d$ ($d \geq 2$)
with mean drift that is asymptotically zero. Specifically, if the mean drift at
$\bx \in \Z^d$ is of magnitude $O(\| \bx\|^{-1})$, we show that $\tau<\infty$
a.s. for any cone. On the other hand, for an appropriate drift field with mean
drifts of magnitude $\| \bx\|^{-\beta}$, $\beta \in (0,1)$, 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.
