Probabilities of large values for sums of i.i.d. non-negative random variables with regular tail of index $-1$

TL;DR

This paper investigates the asymptotic behavior of the probability that the sum of i.i.d. non-negative variables with regularly varying tail of index -1 exceeds a large threshold, comparing it to the maximum's tail probability.

Contribution

It characterizes when the sum's large deviations are dominated by the maximum and introduces correction terms involving the integrated tail for other regimes.

Findings

Identifies regimes where sum and maximum tail probabilities are asymptotically equivalent.

Provides correction terms involving the integrated tail for non-dominant regimes.

Clarifies the asymptotic behavior of sums with regularly varying tails of index -1.

Abstract

Let $\xi_1, \xi_2, \dots$ be i.i.d. non-negative random variables whose tail varies regularly with index $-1$, let $S_n$ be the sum and $M_n$ the largest of the first $n$ values. We clarify for which sequences $x_n\to\infty$ we have $\mathbb P(S_n \ge x_n) \sim \mathbb P(M_n \ge x_n)$ as $n\to\infty$. Outside this regime, the typical size of $S_n$ conditioned on exceeding $x_n$ is not completely determined by the largest summand and we provide an appropriate correction term which involves the integrated tail of $\xi_1$.