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 are not regularly varying and the sequence 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 with a.s. error O (√(logn)1+ε) and, surprisingly, the error term is asymptotically normal, exhibiting an unexpected central limit theorem in St. Petersburg theory.
Originalsprache | englisch |
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Seiten (von - bis) | 4500-4509 |
Seitenumfang | 10 |
Fachzeitschrift | Stochastic Processes and their Applications |
Jahrgang | 129 |
Ausgabenummer | 11 |
DOIs | |
Publikationsstatus | Veröffentlicht - Nov. 2019 |
Schlagwörter
- St. Petersburg sums
- semistable process
- strong approximation
- central limit theorem
ASJC Scopus subject areas
- Statistik und Wahrscheinlichkeit
- Modellierung und Simulation
- Angewandte Mathematik