## The Annals of Probability

- Ann. Probab.
- Volume 19, Number 4 (1991), 1737-1755.

### Strong Limit Theorems of Empirical Functionals for Large Exceedances of Partial Sums of I.I.D. Variables

#### Abstract

Let $(X_i,U_i)$ be pairs of i.i.d. bounded real-valued random variables ($X_i$ and $U_i$ are generally mutually dependent). Assume $E\lbrack X_i\rbrack < 0$ and $\Pr\{X_i > 0\} > 0$. For the (rare) partial sum segments where $\sum^l_{i=k}X_i \rightarrow \infty$, strong limit laws are derived for the sums $\sum^l_{i=k}U_i$. In particular a strong law for the length $(l - k + 1)$ and the empirical distribution of $U_i$ in the event of large segmental sums of $\sum X_i$ are obtained. Applications are given in characterizing the composition of high scoring segments in letter sequences and for evaluating statistical hypotheses of sudden change points in engineering systems.

#### Article information

**Source**

Ann. Probab., Volume 19, Number 4 (1991), 1737-1755.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176990232

**Digital Object Identifier**

doi:10.1214/aop/1176990232

**Mathematical Reviews number (MathSciNet)**

MR1127724

**Zentralblatt MATH identifier**

0746.60028

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F15: Strong theorems

Secondary: 60F10: Large deviations 60G50: Sums of independent random variables; random walks

**Keywords**

Strong laws large segmental sums empirical functionals

#### Citation

Dembo, Amir; Karlin, Samuel. Strong Limit Theorems of Empirical Functionals for Large Exceedances of Partial Sums of I.I.D. Variables. Ann. Probab. 19 (1991), no. 4, 1737--1755. doi:10.1214/aop/1176990232. https://projecteuclid.org/euclid.aop/1176990232