Brazilian Journal of Probability and Statistics

Dispersion models for geometric sums

Bent Jørgensen and Célestin C. Kokonendji

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Abstract

A new class of geometric dispersion models associated with geometric sums is introduced by combining a geometric tilting operation with geometric compounding, in much the same way that exponential dispersion models combine exponential tilting and convolution. The construction is based on a geometric cumulant function which characterizes the geometric compounding operation additively. The so-called v-function is shown to be a useful characterization and convergence tool for geometric dispersion models, similar to the variance function for natural exponential families. A new proof of Rényi’s theorem on convergence of geometric sums to the exponential distribution is obtained, based on convergence of v-functions. It is shown that power v-functions correspond to a class of geometric Tweedie models that appear as limiting distributions in a convergence theorem for geometric dispersion models with power asymptotic v-functions. Geometric Tweedie models include geometric tiltings of Laplace, Mittag-Leffler and geometric extreme stable distributions, along with geometric versions of the gamma, Poisson and gamma compound Poisson distributions.

Article information

Source
Braz. J. Probab. Stat. Volume 25, Number 3 (2011), 263-293.

Dates
First available in Project Euclid: 22 August 2011

Permanent link to this document
http://projecteuclid.org/euclid.bjps/1313973395

Digital Object Identifier
doi:10.1214/10-BJPS136

Mathematical Reviews number (MathSciNet)
MR2832887

Zentralblatt MATH identifier
1271.62025

Keywords
Geometric compounding geometric cumulants geometric infinite divisibility geometric tilting geometric Tweedie model v-function

Citation

Jørgensen, Bent; Kokonendji, Célestin C. Dispersion models for geometric sums. Braz. J. Probab. Stat. 25 (2011), no. 3, 263--293. doi:10.1214/10-BJPS136. http://projecteuclid.org/euclid.bjps/1313973395.


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