Limit theorems for nonnegative independent random variables with truncation

We investigate asymptotic behavior of sums of independent and truncated random variables specified by P (0 ≦X<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\leqq X < \infty}$$\end{document}) = 1 and P (X>x)≍x-α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(X > x) \asymp x^{-\alpha }}$$\end{document} for α>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\alpha > 0}$$\end{document}. By varying truncation levels we study strong laws of large numbers and central limit theorems. These are extensions of the results of Győrfi and Kevei [12] concerning the St. Petersburg game.