On the rate of convergence of the St. Petersburg game
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ON THE RATE OF CONVERGENCE OF THE ST. PETERSBURG GAME ´szlo ´ Gyo ¨ rfi1 and Pe ´ter Kevei2 La 1
Department of Computer Science and Information Theory Budapest University of Technology and Economics Magyar Tud´ osok k¨ or´ utja 2, Budapest, Hungary, H-1117 E-mail: [email protected] 2
Analysis and Stochastics Research Group Hungarian Academy of Sciences Aradi v´ertan´ uk tere 1, Szeged, Hungary, H-6720 E-mail: [email protected]; and Centro de Investigaci´ on en Matem´ aticas, Callej´ on Jalisco S/N, Mineral de Valenciana, Guanajuato 36240, Mexico (Received March 3, 2010; Accepted August 31, 2010)
Dedicated to Endre Cs´ aki and P´ al R´ev´esz on the occasion of their 75th birthdays
Abstract We investigate the repeated and sequential portfolio St. Petersburg games. For the repeated St. Petersburg game, we show an upper bound on the tail distribution, which implies a strong law for a truncation. Moreover, we consider the problem of limit distribution. For the sequential portfolio St. Petersburg game, we obtain tight asymptotic results for the growth rate of the game.
1. Introduction Consider the simple St. Petersburg game, where the player invests 1$ and a fair coin is tossed until a tail first appears, ending the game. If the first tail appears in step k then the payoff X is 2k and the probability of this event is 2−k : P{X = 2k } = 2−k .
(1)
Mathematics subject classification number : 60E05, 60F15, 60G50. Key words and phrases: St. Petersburg games, truncation, almost sure properties, limit distribution, portfolio games. The work was supported in part by the Computer and Automation Research Institute of the Hungarian Academy of Sciences and by the PASCAL2 Network of Excellence under EC grant no. 216886 and by the Hungarian Scientific Research Fund, Grant T-048360. 0031-5303/2011/$20.00
c Akad´emiai Kiad´o, Budapest
Akad´ emiai Kiad´ o, Budapest Springer, Dordrecht
14
¨ L. GYORFI and P. KEVEI
The distribution function of the gain is F (x) = P{X ≤ x} =
0, 1−
1 2⌊log2 x⌋
=1−
2{log2 x} , x
if x < 2, if x ≥ 2,
(2)
where ⌊x⌋ is the usual integer part of x, {x} stands for the fractional part and log2 denotes the logarithm with base 2. Since E{X} = ∞, this game has delicate properties (cf. Bernoulli [2]). In the literature, usually the repeated St. Petersburg game (called iterated St. Petersburg game, too) means multi-period game such that it is a sequence of simple St. Petersburg games, where in each round the player invests 1$. Let Xn denote the payoff for the n-th simple game. Assume that the sequence {Xn }∞ n=1 is i.i.d. After n rounds Pn the player’s gain in the repeated game is i=1 Xi ; Feller [12] proved that lim
n→∞
Pn
i=1 Xi =1 n log2 n
in probability. In Section 2 we revisit the a.s. properties of the repeated St. Petersburg game, in Section 3 we investigate the limit distributions of the truncated sums under different truncation levels. This analysis allow us to understand ‘where the important things happen’. In Section 4 we show the consequences for sequential portfolio games with
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