Open Access
December, 1977 A Counterexample in the Approximation Theory of Random Summation
Dieter Landers, Lothar Rogge
Ann. Probab. 5(6): 1018-1023 (December, 1977). DOI: 10.1214/aop/1176995669


Let $X_n, n \in \mathbb{N}$, be independent and identically distributed random variables and $\tau_n$ be random summation indices such that $\tau_n/n \rightarrow \tau > 0$ in probability. It is shown that even if $\tau_n/n$ converges to $\tau$ as quickly as possible (i.e., $\tau_n/n = \tau$) no general approximation orders for suitably normalized random sums $\sum^{\tau_n(\omega)}_{\nu=1} X_\nu(\omega)$ are available. If, however, the limit function $\tau$ is independent of $X_n, n \in \mathbb{N}$, we give a positive approximation result.


Download Citation

Dieter Landers. Lothar Rogge. "A Counterexample in the Approximation Theory of Random Summation." Ann. Probab. 5 (6) 1018 - 1023, December, 1977.


Published: December, 1977
First available in Project Euclid: 19 April 2007

zbMATH: 0373.60028
MathSciNet: MR455079
Digital Object Identifier: 10.1214/aop/1176995669

Primary: 60F05
Secondary: 60G50

Keywords: convergence order , Independent random variables , independent summation index , summation index depending on summands

Rights: Copyright © 1977 Institute of Mathematical Statistics

Vol.5 • No. 6 • December, 1977
Back to Top