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

For a sequence of random variables (X1, X2, . . . , Xn), n ≥ 1, that are independent and identically distributed with a regularly varying tail with index −α, α ≥ 0, we show that the contribution of the maximum term Mn , max(X1, . . . , Xn) in the sum Sn , X1 + · · · + Xn, as n → ∞, decreases monotonically with α in stochastic ordering sense.