A strong law of large numbers for trimmed sums, with applications to generalized St. Petersburg games
type: Item
instance of: scholarly article
title: A strong law of large numbers for trimmed sums, with applications to generalized St. Petersburg games
zbMATH Open document ID: 0859.60030
DOI: 10.1016/0167-7152(94)00253-3
author: Gordon Simons
author: Sándor Csörgő
published in: Statistics \& Probability Letters
publication date: 1996-07-31T00:00:00Z
review text: For \(0<p<1\) and \(0<\alpha<2\) with \(q=1-p\) consider a generalized St. Petersburg game in which the gain \(X\) of the player is such that \(P(X=q^{-k/\alpha})=q^{k-1}p\), \(k=1,2,\dots\). The classical St. Petersburg game corresponds to \(\alpha=1\) and \(p=1/2\). The distribution function \(F\) of \(X\) satisfies \[ 1-F(x)=q^{[\alpha\log_{1/q}x]}=l(x)/x^\alpha,\quad x\geq 1, \] where \([y]=\max\{k\in\mathbb{Z}:k\leq y\}\) and \(\log_{1/q}u\) is the logarithm of \(u>0\) to the base \(1/q\). The function \(l\) satisfies \(1\leq l(x)\leq 1/q\) for all \(x\geq 1\) but, due to its oscillating behavior, is not slowly varying at infinity, though it is ``sandwiched'' by the two slowly varying (in fact, constant) functions 1 and \(1/q\). The authors show that a characterization of the almost sure asymptotic stability of lightly trimmed sums of upper order statistics when the right tail of the underlying distribution with positive support is regularly varying (i.e. \(l\) is slowly varying) which has been obtained by \textit{J. H. J. Einmahl}, \textit{D. M. Mason} and the reviewer [Probab. Theory Relat. Fields 79, No. 1, 59-74 (1988; Zbl 0629.62019)] can be extended to this situation. Consequently, this more general result is then applicable to the cumulative gains in a sequence of independent generalized St. Petersburg games, yielding additional insight into the asymptotic behavior of this famous paradoxon in probability theory.
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 62G30
zbMATH DE Number: 885135
zbMATH Keywords: lightly trimmed sums of order statistics
zbMATH Keywords: almost sure asymptotic stability
zbMATH Keywords: generalized St. Petersburg games
MaRDI profile type: Publication
full work available at URL: 0167-7152(94)00253-3
OpenAlex ID: W1541271516
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description: scientific article; zbMATH DE number 885135