On a zero-drift nearest-neighbour random walk

The present study concerns the analysis of the hitting point identity for a nearest-neighbour random walk of which the one-step transition to the $NE$, $SE$, $SW$ and $NW$ are the only transitions with nonzero probabilities. The one-step transition vector has a symmetrical probability distribution with zero drifts. The state space of the random walk is the set of lattice points in the first quarter plane, the point at the coordinate axes are all absorbing states. The distribution of the hitting point with the axes is investigated for the case $-1< rho < 0$ and for the case $rho =0$, here $rho$ is the correlation of the components of the one-step transition vector. For $-1< rho < 0$ the generating function of this distribution is derived. For $rho =0$ the distribution is calculated explicitly.