Parameter-dependent renewal theorems with applications

Abstract

Letχ 1, χ2, ... be a sequence of i.i.d. random variables with positive mean\(\tilde \mu \) and finite variance\(\tilde \sigma ^2 \) and letr(b), b⩾0, be real numbers tending to 0 asb → ∞. Definings n=χ1+...+χn andS n=Sn(b)=sn+r(b)n, the stopping time τ=τ(b)=inf {n>/1:Sn >b} whereb=b(b) → ∞, will be considered with special regard to the excess over the boundaryR b=sτ+r(b)τ−b. It turns out that the limiting distribution ofR b is the same as in the caser(b)≡0 for allb. Proving this, Blackwell's renewal theorem and its integral version have to be established first in the above stated situation. Finally, an expansion ofEτ to vanishing terms asb→∞ will be provided and applied to some examples arising in economics.