## The Annals of Probability

### 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

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