The Annals of Probability

Tight Bounds for the Renewal Function of a Random Walk

D. J. Daley

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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.

Article information

Ann. Probab., Volume 8, Number 3 (1980), 615-621.

First available in Project Euclid: 19 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60K05: Renewal theory
Secondary: 60K25: Queueing theory [See also 68M20, 90B22]

Renewal function bounds ladder variables


Daley, D. J. Tight Bounds for the Renewal Function of a Random Walk. Ann. Probab. 8 (1980), no. 3, 615--621. doi:10.1214/aop/1176994732.

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