Limit theorems for multidimensional renewal sets

TL;DR

This paper establishes limit theorems for multidimensional renewal sets derived from sums of i.i.d. variables, extending classical results to the set-valued context with convergence characterized by set distances.

Contribution

It introduces strong law, law of the iterated logarithm, and distributional limit theorems for multidimensional renewal sets, a novel set-based perspective in renewal theory.

Findings

Proves strong law of large numbers for renewal sets

Establishes law of the iterated logarithm for set convergence

Derives distributional limit theorem for the set-valued process

Abstract

Consider multiple sums $S_n$ on the $d$-dimensional integer grid,which are generated by i.i.d.\ random variables with a positive expectation. We prove the strong law of large numbers, the law of the iterated logarithm and the distributional limit theorem for random sets ${\mathcal M}_t$ that appear as inversion of the multiple sums, that is, as the set of all arguments $x\in{\mathbb R}_+^d$ such that the interpolated multiple sum $S_x$ exceeds $t$. The moment conditions are identical to those imposed in the almost sure limit theorems for multiple sums. The results are expressed in terms of set inclusions and using distances between sets.