Tight Bounds for the Renewal Function of a Random Walk

Abstract

It is shown that for a random walk $\{S_n\}$ starting at the origin having generic step random variable $X$ with finite second moment and positive mean $\lambda^{-1} = EX$, the renewal function $U(y) = E {\tt\#}\{n = 0,1, \cdots: S_n \leqslant y\}$ satisfies for $y \geqslant 0$ $$|U(y) - \lambda y - \frac{1}{2}\lambda^2EX^2| \leqslant \frac{1}{2}\lambda^2EX^2 - \lambda EM \leqslant \frac{1}{2}\lambda^2EX^2_+$$ where $M = - \inf_{n\geqslant 0}S_n$. Both the upper and lower bounds are attained by simple random walk. Bounds are also given for $U(-y)(y \geqslant 0)$ and for the renewal function of a transient renewal process when $\Pr\{X \geqslant 0\} = 1 > \Pr\{0 \leqslant X < \infty\}$. The proof uses a Wiener-Hopf like identity relating $U$ to the renewal functions of the ascending and descending ladder processes to which Lorden's tight bound for the renewal process case is applied.