A renewal density theorem in the multi-dimensional case

Abstract

Summary: In this note a renewal density theorem in the multi-dimensional case is formulated and proved. Let f( x ) be the density function of a p-dimensional random variable with positive mean vector μ and positive-definite covariance matrix Σ, let h_n ( x ) be the n-fold convolution of f( x ) with itself, and set Then for arbitrary choice of integers k ₁, …, k_{p– 1} distinct or not in the set (1, 2, …, p), it is shown that under certain conditions as all elements in the vector x = (x ₁, …, x_p ) become large. In the above expression μ‵ is interpreted as a row vector and μ a column vector. An application to the theory of a class of age-dependent branching processes is also presented.