Probabilities of large values for sums of i.i.d. non-negative random variables with regular tail 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 ${\xi }_1$.

14 pages