On some first-crossing-time probabilities for a two-dimensional random walk with correlated components

For a two-dimensional random walk {X (n) = (X(n)1, X(n)2)T, n . .0} with correlated components the first-crossing-time probability problem through unit-slope straight lines x2 = x1 - r(r = 0,1) is analysed. The p.g.f.'s for the first-crossing-time probabilities are expressed as solutions of a fourth-degree algebraic equation and are then exploited to obtain the first-crossing-time probabilities. Several additional results, including the mean first-crossing time and the probability of ultimate crossing, are also given.