We consider a simple random walk on $\mathbb{Z}^d$ started at the origin and stopped on its first exit time from $({-}L,L)^d \cap \mathbb{Z}^d$. Write L in the form $L = m N$ with $m = m(N)$ and N an integer going to infinity in such a way that $L^2 \sim A N^d$ for some real constant $A \gt 0$. Our main result is that for $d \ge 3$, the projection of the stopped trajectory to the N-torus locally converges, away from the origin, to an interlacement process at level $A d \sigma_1$, where $\sigma_1$ is the exit time of a Brownian motion from the unit cube $({-}1,1)^d$ that is independent of the interlacement process. The above problem is a variation on results of Windisch (2008) and Sznitman (2009).

Consider a sum ∑1NYi of random variables conditioned on a given value of the sum ∑1NXi of some other variables, where Xi and Yi are dependent but the pairs (Xi,Yi) form an i.i.d. sequence. We consider here the case when each Xi is discrete. We prove, for a triangular array ((Xni,Yni)) of such pairs satisfying certain conditions, both convergence of the distribution of the conditioned sum (after suitable normalization) to a normal distribution, and convergence of its moments. The results are motivated by an application to hashing with linear probing; we give also some other applications to occupancy problems, random forests, and branching processes.

Let (X1, X2, · ··, Xn) be a vector of discrete splittable and interchangeable random variables, and let Sj = Σji = 1Xi. This paper is concerned with probabilities of the form P{Sj < jb – d for j = 1, 2, 3, ···, n | Sn = r}. We focus on applications of these probabilities, study their properties and suggest how they may be computed. The applications are from the areas of computer science, economics, operations management, finance and risk theory.