Renewal theory in two dimensions: asymptotic results

Abstract

In an earlier paper (Renewal theory in two dimensions: Basic results) the author developed a unified theory for the study of bivariate renewal processes. In contrast to this aforementioned work where explicit expressions were obtained, we develop some asymptotic results concerning the joint distribution of the bivariate renewal counting process (N _x(1), N _y(2)), the distribution of the two-dimensional renewal counting process N _x,y and the two-dimensional renewal function &N _x,y. A by-product of the investigation is the study of the distribution and moments of the minimum of two correlated normal random variables. A comprehensive bibliography on multi-dimensional renewal theory is also appended.