## The Annals of Probability

- Ann. Probab.
- Volume 10, Number 4 (1982), 935-941.

### Moments and Error Rates of Two-Sided Stopping Rules

#### Abstract

For $X_1, X_2,\cdots$ i.i.d., $EX_1 = \mu \neq 0, S_n = X_1 + \cdots + X_n$, the asymptotic behavior of moments and error rates of the two-sided stopping rules $\inf \{n \geq 1: |S_n| > cn^\alpha\}, c > 0, 0 \leq \alpha < 1$, is considered. Convergence of (normalized) moments of all orders as $c \rightarrow \infty$ is obtained, without the higher moment assumptions needed in the one-sided case of extended renewal theory (Gut, 1974), and in a more general setting than just the i.i.d. case. Necessary and sufficient conditions are given for convergence of series involving the error rates, in terms of the moments of $X_1$.

#### Article information

**Source**

Ann. Probab., Volume 10, Number 4 (1982), 935-941.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176993715

**Digital Object Identifier**

doi:10.1214/aop/1176993715

**Mathematical Reviews number (MathSciNet)**

MR672294

**Zentralblatt MATH identifier**

0496.60040

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]

Secondary: 60G50: Sums of independent random variables; random walks 62L10: Sequential analysis

**Keywords**

Stopping rules uniform integrability moment convergence delayed sums error rates of sequential tests

#### Citation

Martinsek, Adam T. Moments and Error Rates of Two-Sided Stopping Rules. Ann. Probab. 10 (1982), no. 4, 935--941. doi:10.1214/aop/1176993715. https://projecteuclid.org/euclid.aop/1176993715