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

TL;DR

This paper investigates how the largest term influences the sum of i.i.d. random variables with regularly varying tails, revealing that its contribution diminishes as the tail index increases.

Contribution

It provides a stochastic ordering analysis of the maximum's contribution to the sum for variables with regularly varying tails, highlighting the effect of tail index on this contribution.

Findings

Maximum contribution decreases with increasing tail index

Contribution diminishes monotonically in stochastic order

Results apply to sums of i.i.d. variables with regularly varying tails

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.