A strong law of large numbers for trimmed sums, with applications to generalized St. Petersburg games
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.
lightly trimmed sums of order statistics
scientific article; zbMATH DE number 885135
scientific article