Limit theorems for multidimensional renewal sets

Abstract

Consider the multiple sums Sn of i.i.d. random variables with a positive expectation on the d-dimensional integer grid. We prove the strong law of large numbers, the law of the iterated logarithm and the distributional limit theorem for random sets which appear as upper excursion sets of the interpolated multiple sums, that is, as the set of all arguments $${x\in\mathbb{R}_+^d}$$ such that the interpolated multiple sums Sx exceed t. The results are expressed in terms of set inclusions and using distances between sets.