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June, 1978 Asymptotic Behaviour of the Variance of Renewal Processes and Random Walks
D. J. Daley, N. R. Mohan
Ann. Probab. 6(3): 516-521 (June, 1978). DOI: 10.1214/aop/1176995536


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 $B$.


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D. J. Daley. N. R. Mohan. "Asymptotic Behaviour of the Variance of Renewal Processes and Random Walks." Ann. Probab. 6 (3) 516 - 521, June, 1978.


Published: June, 1978
First available in Project Euclid: 19 April 2007

zbMATH: 0378.60068
MathSciNet: MR474534
Digital Object Identifier: 10.1214/aop/1176995536

Primary: 60K05
Secondary: 60J15

Keywords: asymptotic variance , Random walk , Renewal theorem

Rights: Copyright © 1978 Institute of Mathematical Statistics

Vol.6 • No. 3 • June, 1978
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