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

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$n \to \infty$,decreasesmonotonicallywith$\alpha$ in stochastic ordering sense.