Random Walk with Zero Drifts in the First Quadrant of R2

摘要

The random walk on the lattice in the positive quadrant is studied for the casethat the one step displacement vector has zero drift, finite second moments and support contained in the set whose members are -1, 0, 1, 0 and so on by the set -1, 0, 1, 2 and so on. It is well known that the first entrance time out from a point in the interior of the lattice into union of the coordinate axes is finite probability one. It is shown that its first moment is finite if and only if the covariance of the x and y component of the one step displacement vector is negative. For this case explicit expressions are given for the first moment of the entrance time and for the first and second moments of the hitting point of the axes, in terms of the second moment characteristics of the one step displacement vector. The results are deduced from the hitting point identity for the random walk.