Contribution of the Extreme Term in the Sum of Samples with Regularly Varying Tail

Abstract

For a sequence of random variables $(X_1,X_2,\dots ,X_n)$, $n\ge 1$, that are independent and identically distributed with a regularly varying tail with index $-\alpha$, $\alpha \ge 0$, we show that the contribution of the maximum term $M_n\triangleq \max (X_1,\dots ,X_n)$ in the sum $S_n\triangleq X_1+\cdots +X_n$, as $n\to \infty$, decreases monotonically with $\alpha$ in stochastic ordering sense.

9 pages, submitted for possible publication