First-crossing and ballot-type results for some nonstationary sequences

In this paper we consider the problem of first-crossing from above for a partial sums process m+St, t . 1, with the diagonal line when the random variables Xt, t . 1, are independent but satisfying nonstationary laws. Specifically, the distributions of all the Xts belong to a common parametric family of arithmetic distributions, and this family of laws is assumed to be stable by convolution. The key result is that the first-crossing time distribution and the associated ballot-type formula rely on an underlying polynomial structure, called the generalized Abel-Gontcharoff structure. In practice, this property advantageously provides simple and efficient recursions for the numerical evaluation of the probabilities of interest. Several applications are then presented, for constant and variable parameters.