The Annals of Probability

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

Amir Dembo and Samuel Karlin

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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.


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