**Sankhya: The Indian Journal of Statistics**

1995, Volume 57, Series A, Pt. 1, pp. 118--125

**DETERMINANCY OF MEANS OF ORDER STATISTICS**

By

M. C. SPRUILL, *Georgia Institute of Technology*

SUMMARY. If X_{1}, X_{2},.... are independent random variables distributed as X and if $E[|X|] < \infty$ then the sequence $\rho_n = E[X_1 \vee X_2 \vee ... \vee X_n]$ of maximal moments determines the probability distribution of X. Exploiting the fact that a collection of functions is fundamental if and only if a corresponding moment problem is determinate yields simple proofs of the claim above, a theorem of Pollak on the determinacy of more general sequences of means of order statistics, and that a subsequence n(i) of the maximal moments determines the distribution if and only if $\sum_{i \geq 1} n^{-1}(i)=\infty$.

*AMS (1990) subject classification.* Primary 62E10, secondary 41A10.

*Key words and phrases.* Fundamental sets, Müntz theorems, moments.