Strong approximation and a central limit theorem for St. Petersburg sums

Berkes, I. (2018) Strong approximation and a central limit theorem for St. Petersburg sums. STOCHASTIC PROCESSES AND THEIR APPLICATIONS. pp. 1-11. ISSN 0304-4149

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Abstract

The St. Petersburg paradox (Bernoulli 1738) concerns the fair entry fee in a game where the winnings are distributed as P(X = 2k) = 2−k, k = 1, 2, . . .. The tails of X are not regularly varying and the sequence Sn of accumulated gains has, suitably centered and normalized, a class of semistable laws as subsequential limit distributions (Martin-L¨of (1985), Cs¨org˝o and Dodunekova (1991)). This has led to a clarification of the paradox and an interesting and unusual asymptotic theory in past decades. In this paper we prove that Sn can be approximated by a semistable L´evy process {L(n), n ≥ 1} with a.s. error O( √ n(log n)1+ε) and, surprisingly, the error term is asymptotically normal, exhibiting an unexpected central limit theorem in St. Petersburg theory.

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