Tail probabilities of St. Petersburg sums, trimmed sums, and their limit

TL;DR

This paper derives precise asymptotic tail probabilities for trimmed sums of St. Petersburg random variables, revealing near-subexponential behavior and characterizing the tail limits.

Contribution

It provides exact asymptotics for tail probabilities of trimmed sums of St. Petersburg variables and clarifies their subexponential-like properties.

Findings

Exact tail asymptotics for fixed n as x→∞

Near-subexponential tail behavior of the distribution

Characterization of tail limits for trimmed sums

Abstract

We provide exact asymptotics for the tail probabilities $\mathbb{P} \{S_{n,r} > x\}$ as $x \to \infty$, for fix $n$, where $S_{n,r}$ is the $r$-trimmed partial sum of i.i.d. St. Petersburg random variables. In particular, we prove that although the St. Petersburg distribution is only O-subexponential, the subexponential property almost holds. We also determine the exact tail behavior of the $r$-trimmed limits.