Tail Probabilities of St. Petersburg Sums, Trimmed Sums, and Their Limit

István Berkes,Péter Kevei,László Györfi

doi:10.1007/s10959-016-0677-5

Abstract

We provide exact asymptotics for the tail probabilities $${\mathbb {P}}\{ S_{n,r} > x \}$$ as $$x \rightarrow \infty $$ , for fixed 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.

Highlights

Peter offers to let Paul toss a fair coin repeatedly until it lands heads and pays him 2k ducats if this happens game
As a continuation of our studies of the joint behavior of Sn and Xn∗ in [9], we investigate the properties of the trimmed sum Sn − Xn∗ both for fix n and for n → ∞
In Theorem 2 we show that P{(Sn − Xn∗)/n − log2 n > x} ≤

Summary

Introduction

Peter offers to let Paul toss a fair coin repeatedly until it lands heads and pays him 2k ducats if this happens game. The limit theorems (3) and (4) suggest that the irregular oscillating behavior is due to the maximum, which is indicated by the following fact. In Theorem 3 we determine {Wγ∗ : γ ∈ (1/2, 1]}, the set of the possible subsequential limit distributions of (Sn − Xn∗)/n − log n. This result was first obtained by Gut and Martin-Lof in their Theorem 6.1 in [12].

Preliminaries

Uniform tail bound for the trimmed sums

Properties of the 1-trimmed limit

Representation of the r-trimmed limit