Electronic Communications in Probability

The $m(n)$ out of $k(n)$ bootstrap for partial sums of St. Petersburg type games

Eustasio del Barrio, Arnold Janssen, and Markus Pauly

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This paper illustrates that the bootstrap of a partial sum given by i.i.d. copies of a random variable $X_1$ has to be dealt with care in general. It turns out that in various cases a whole spectrum of different limit laws of the $m(n)$ out of $k(n)$ bootstrap can be obtained for different choices of $m(n)/k(n) -> 0$ whenever $X_1$ does not lie in the domain of attraction of a stable law. As a concrete example we study bootstrap limit laws for the cumulated gain sequence of repeated St. Petersburg games. It is shown that here a continuum of different semi-stable bootstrap limit laws occurs. <br />

Article information

Electron. Commun. Probab., Volume 18 (2013), paper no. 91, 10 pp.

Accepted: 3 December 2013
First available in Project Euclid: 7 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60E07: Infinitely divisible distributions; stable distributions
Secondary: 62F40: Bootstrap, jackknife and other resampling methods 60F05: Central limit and other weak theorems

Bootstrap infinitely divisible distributions L{\'e}vy process $m(n)$ out of $k(n)$ resampling stable and semi-stable laws St. Petersburg game

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del Barrio, Eustasio; Janssen, Arnold; Pauly, Markus. The $m(n)$ out of $k(n)$ bootstrap for partial sums of St. Petersburg type games. Electron. Commun. Probab. 18 (2013), paper no. 91, 10 pp. doi:10.1214/ECP.v18-2772. https://projecteuclid.org/euclid.ecp/1465315630

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