Asymptotic behavior of the generalized St. Petersburg sum conditioned on its maximum

In this paper, we revisit the classical results on the generalized St. Petersburg sums. We determine the limit distribution of the St. Petersburg sum conditioning on its maximum, and we analyze how the limit depends on the value of the maximum. As an application, we obtain an infinite sum representa...

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Bibliographic Details
Published in:Bernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability 2016-05, Vol.22 (2), p.1026-1054
Main Authors: FUKKER, GÁBOR, GYÖRFI, LÁSZLÓ, KEVEI, PÉTER
Format: Article
Language:English
Subjects:
conditional limit theorem
generalized St. Petersburg distribution
merging theorem
semistable law
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Summary:In this paper, we revisit the classical results on the generalized St. Petersburg sums. We determine the limit distribution of the St. Petersburg sum conditioning on its maximum, and we analyze how the limit depends on the value of the maximum. As an application, we obtain an infinite sum representation of the distribution function of the possible semistable limits. In the representation, each term corresponds to a given maximum, in particular this result explains that the semistable behavior is caused by the typical values of the maximum.
ISSN:1350-7265
DOI:10.3150/14-BEJ685