Generalized one-sided laws of the iterated logarithm for random variables barely with or without finite mean

Large gains in the St. Petersburg game

“…It was subsequently shown that, with probability one, the set of limit points of S n /(n log 2 n) is the interval [1, ∞), cf. Chow and Robbins [2] and Adler [1]. This result was refined by Martin-Löf [9] and Csörgő-Dodunekova [3], who identified the class of subsequential distributional limits of S n /n − log 2 n, and by Csörgő and Simons [4] who showed that, with probability one, S n is asymptotic to n log 2 n if the largest gains are ignored (i.e., the entry fee is fair except for the largest gains).…”

“…All these limit laws suggest finding the deviation rates of the accumulated gain S n in the St. Petersburg game. Hu and Nyrhinen [8] and Gantert [6] obtained the following result for the polynomial size gains, that easily implies the deviation rate for Feller's normalization, or the iterated logarithm normalization, as in Adler [1] or Vardi [11]: for ε > 0 and b > 1, one has lim n→∞ log 2 P (S n > εx n )…”

“…Hence, if X is the gain at a single trial, we have P X = 2 k = 2 −k and P (X > c) = 2 −[log 2 c] , (1) for k = 1, 2, . .…”

Exact Weak Laws and One Sided Strong Laws

“…The easy part was done in both [2] and [5]. By using the technique from [1] we can get an exact lower bound. But the problem is in selecting the right sequence d n .…”

The St. Petersburg Paradox and the Crash of High-Tech Stocks in 2000

“…We refer to Adler (1990), Berkes, Csáki, and Csörgő (1999), Csörgő and Simons (1996), and Csörgő (2003) for recent developments in this area and for further references. Especially intriguing is the remarkable article by Csörgő and Simons (2002) which reveals new paradoxes within the structure of the classical St. Petersburg paradox; in particular, their results show that, paradoxically, there may be a very different outcome if n distinct Pauls play one St. Petersburg game each than if one Paul plays n games.…”