Transformations of the simplest nonsymmetric random walks and some applications of the invariance principle

We derive convenient formulas for the tail probabilities of the supnorm of the simplest nonsymmetric random walks defined on a finite time-interval. Using these formulas, we obtain a new representation for the distribution of the number of crossings of a canonical strip by the random walks. As a consequence of the above-mentioned results, we propose a new approach to calculation of the distributions of some boundary functionals of a Wiener process with drift.