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-Löf, 1985; Csörgő 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 Lévy process {L(n),n≥1} with a.s. error O(√n(logn)1+ε) and, surprisingly, the error term is asymptotically normal, exhibiting an unexpected central limit theorem in St. Petersburg theory.
Original language | English |
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Pages (from-to) | 4500-4509 |
Number of pages | 10 |
Journal | Stochastic Processes and their Applications |
Volume | 129 |
Issue number | 11 |
DOIs | |
Publication status | Published - Nov 2019 |
Keywords
- Central limit theorem
- Semistable process
- St. Petersburg sums
- Strong approximation
ASJC Scopus subject areas
- Statistics and Probability
- Modelling and Simulation
- Applied Mathematics