A multi-dimensional renewal theorem for finite Markov chains
Let U, L and F be functions from Z d into the set of real square matrices of finite dimension N, and let in addition L(t) be positive for each t. Define the convolution L*U by the formula L*U(t)=Σ t1 + t2 = t L(t 1 )U(t 2 ), and put R=Σ ∞ n=0 L n* , provided the sum converges. Here L 0* =δ, where δ(0)=1 (the identity matrix) and δ(t)=0 for t¬=0, and L n* =L*L (n−1)* for n≥1. A solution U of the renewal equation U-L*U=F is then given by U=R*F, provided the latter expression converges. The object of the paper is to study the asymptotic behaviour of R*F(t), as |t|→∞. The result can be applied to first passage problems for sums of Markov dependent random variables