Abstract. We show that addition mod 2 n is CCZ-equivalent to a quadratic vectorial Boolean function. We use this to reduce the solution of systems differential equations of addition to the solution of a system of linear equations and to derive a fully explicit formula for the correlation coefficients, which leads to new results about the Walsh transform of addition mod 2 n . The results have applications in the cryptanalysis of cyptographic primitives which use addition mod 2 n .

We study in this paper a generalized coupon collector problem, which consists in analyzing the time needed to collect a given number of distinct coupons that are drawn from a set of coupons with an arbitrary probability distribution. We suppose that a special coupon called the null coupon can be drawn but never belongs to any collection. In this context, we prove that the almost uniform distribution, for which all the non-null coupons have the same drawing probability, is the distribution which stochastically minimizes the time needed to collect a fixed number of distinct coupons. Moreover, we show that in a given closed subset of probability distributions, the distribution with all its entries, but one, equal to the smallest possible value is the one, which stochastically maximizes the time needed to collect a fixed number of distinct coupons. An computer science application shows the utility of these results.

Abstract. We show analogs of the classical arcsine theorem for the occupation time of a random walk in (−∞, 0) in the case of a small positive drift. To study the asymptotic behavior of the total time spent in (−∞, 0) we consider parametrized classes of random walks, where the convergence of the parameter to zero implies the convergence of the drift to zero. We begin with shift families, generated by a centered random walk by adding to each step a shift constant a > 0 and then letting a tend to zero. Then we study families of associated distributions. In all cases we arrive at the same limiting distribution, which is the distribution of the time spent below zero of a standard Brownian motion with drift 1. For shift families this is explained by a functional limit theorem. Using fluctuation-theoretic formulas we derive the generating function of the occupation time in closed form, which provides an alternative approach. In the course also give a new form of the first arcsine law for the Brownian motion with drift.

Abstract. We show that the supremum of the successive percentages of red balls in Pólya's urn model is almost surely rational, give the set of values that are taken with positive probability and derive several exact distributional results for the all-time maximal percentage.

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