**Sankhya:
The Indian Journal of Statistics**

2004, Volume 66, Pt. 3, 428--439

**On Finite Sequences Conditionally
Uniform Given Minima and Maxima**

By

Pilar Iglesias Z., PUC-Santiago,
Chile

Franti\v{s}ek Mat\'{u}\v{s}, Academy of Sciences, Czech Republic

Carlos A.B. Pereira and Nelson I. Tanaka, IME/USP, Brazil

SUMMARY. Conditionings in a finite sequence $X^{(N)} = (X_1 , X_2 , \ldots , X_N)$ of real random variables by $\max X^{(N)}$ and by $\max X^{(N)}$ together with $\min X^{(N)}$ are considered. If $X^{(N)}$ is conditionally uniform in a very general sense with respect to a reference Borel measure $\nu$ then a shorter subsequence $X^{(n)} = (X_1, X_2 ,\ldots , X_n)$, $1\le n < N$, can be well approximated, in the variation distance, by a mixture of $n$-powers of restrictions of $\nu$. These finite de Finetti type results can be used to obtain integral representations of infinite sequences which have all their finite sub-sequences conditionally uniform.

*AMS (1991) subject classification}. ***Primary 62A05; secondary 62A15**.

*Key words and phrases. *Aggregate measures; exchangeability; finite
forms of de Finetti-type theorems; uniform distribution.