Multivariate weighted renewal functions

Let ( X , Y ) , ( X 1 , Y 1 ) , ( X 2 , Y 2 ) , … denote independent positive random vectors with common distribution function F ( x , y ) = P ( X ⩽ x , Y ⩽ y ) with F ( x , y ) < 1 for all x , y . Based on the X i and the Y j we construct the sum sequences S n 1 and S m 2 respectively. For a double sequence of weighting constants { b ( n , m ) } we associate a weighted renewal function G ( x , y ) defined as G ( x , y ) = ∑ n = 0 ∞ ∑ m = 0 ∞ b ( n , m ) P ( S n 1 ⩽ x , S m 2 ⩽ y ) . The function G ( x , y ) can be expressed in terms of well-known renewal quantities. The main goal of this paper is to study asymptotic properties of G ( x , y ) . In the one-dimensional case such results have been obtained among others by Omey and Teugels [Weighted renewal functions: a hierarchical approach, Adv. in Appl. Probab. 34 (2002) 394–415.] and Alsmeyer [Some relations between harmonic renewal measures and certain first passage times, Statist. Probab. Letters 12 (1991) 19–27; On generalized renewal measures and certain first passage times, Ann. Probab. 20 (1992) 1229–1247]. Here we prove a multivariate version of the elementary renewal theorem and moreover we obtain a rate of convergence result in this elementary renewal theorem. We close the paper with an application and some concluding remarks. For convenience we prove and formulate the results in the two-dimensional case only.