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 $B$.