## The Annals of Probability

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

### Asymptotic Behaviour of the Variance of Renewal Processes and Random Walks

#### 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