Asymptotic Behaviour of the Variance of Renewal Processes and Random Walks

Daley, D. J.; Mohan, N. R.

Ann. Probab., Tome 6 (1978) no. 6, p. 516-521 / Harvested from Project Euclid

Résumé

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 $B$.

Publié le : 1978-06-14
Classification: Random walk, asymptotic variance, renewal theorem, 60K05, 60J15

@article{1176995536,
     author = {Daley, D. J. and Mohan, N. R.},
     title = {Asymptotic Behaviour of the Variance of Renewal Processes and Random Walks},
     journal = {Ann. Probab.},
     volume = {6},
     number = {6},
     year = {1978},
     pages = { 516-521},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1176995536}
}

Daley, D. J.; Mohan, N. R. Asymptotic Behaviour of the Variance of Renewal Processes and Random Walks. Ann. Probab., Tome 6 (1978) no. 6, pp.  516-521. http://gdmltest.u-ga.fr/item/1176995536/