Renewal Theorems for Random Walks with Multi-Dimensional Time

Abstract

In this chapter, we consider the problems on the asymptotic behavior of the renewal function and process constructed from multi-indexed sums of independent identically distributed random variables. The asymptotic behavior depends on the dimension $d$ of the space of multi-indices and differs from the classical case. It is worth mentioning that more complicated questions on the rate of convergence of renewal functions and processes constructed from multi-indexed sums of random variables still have no definite answers, since they depend on the Riemann hypothesis.