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

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