The Annals of Probability

Asymptotic Behaviour of the Variance of Renewal Processes and Random Walks

D. J. Daley and N. R. Mohan

Full-text: Open access

Abstract

For a sequence of independent identically distributed random variables $\{X_n\}, n = 1, 2, \cdots,$ yielding the sums $S_n = X_1 + \cdots + X_n$ let $N(x) = \sharp\{n \geqq 1: S_n \leqq x\}$. Results of Stone and the general renewal equation as treated by Feller are used to prove that under certain conditions on the common distribution function of the $X_n$'s, the variance of $N(x)$ is asymptotically like $Ax + B + o(1)$ as $x\rightarrow\infty$ for specified constants $A$ and $B$.

Article information

Source
Ann. Probab., Volume 6, Number 3 (1978), 516-521.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176995536

Digital Object Identifier
doi:10.1214/aop/1176995536

Mathematical Reviews number (MathSciNet)
MR474534

Zentralblatt MATH identifier
0378.60068

JSTOR
links.jstor.org

Subjects
Primary: 60K05: Renewal theory
Secondary: 60J15

Keywords
Random walk asymptotic variance renewal theorem

Citation

Daley, D. J.; Mohan, N. R. Asymptotic Behaviour of the Variance of Renewal Processes and Random Walks. Ann. Probab. 6 (1978), no. 3, 516--521. doi:10.1214/aop/1176995536. https://projecteuclid.org/euclid.aop/1176995536


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