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A bivariate infinitely divisible distribution with exponential and Mittag-Leffler marginals

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  • Kozubowski, Tomasz J.
  • Meerschaert, Mark M.

Abstract

We introduce a bivariate distribution supported on the first quadrant with exponential, and heavy tailed Mittag-Leffer, marginal distributions. Although this distribution belongs to the class of geometric operator stable laws, it is a rather special case that does not follow their general theory. Our results include the joint density and distribution function, Laplace transform, conditional distributions, joint moments, and tail behavior. We also establish infinite divisibility and stability properties of this model, and clarify its connections with operator stable and geometric operator stable laws.

Suggested Citation

  • Kozubowski, Tomasz J. & Meerschaert, Mark M., 2009. "A bivariate infinitely divisible distribution with exponential and Mittag-Leffler marginals," Statistics & Probability Letters, Elsevier, vol. 79(14), pages 1596-1601, July.
    • Handle: RePEc:eee:stapro:v:79:y:2009:i:14:p:1596-1601
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    References listed on IDEAS

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    1. Weron, Karina & Kotulski, Marcin, 1996. "On the Cole-Cole relaxation function and related Mittag-Leffler distribution," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 232(1), pages 180-188.
    2. Kozubowski, Tomasz J. & Meerschaert, Mark M. & Panorska, Anna K. & Scheffler, Hans-Peter, 2005. "Operator geometric stable laws," Journal of Multivariate Analysis, Elsevier, vol. 92(2), pages 298-323, February.
    3. Kozubowski, Tomasz J. & Panorska, Anna K. & Podgórski, Krzysztof, 2008. "A bivariate Lévy process with negative binomial and gamma marginals," Journal of Multivariate Analysis, Elsevier, vol. 99(7), pages 1418-1437, August.
    4. Kozubowski Tomasz J., 1994. "The Inner Characterization Of Geometric Stable Laws," Statistics & Risk Modeling, De Gruyter, vol. 12(3), pages 307-322, March.
    5. R. Pillai, 1990. "On Mittag-Leffler functions and related distributions," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 42(1), pages 157-161, March.
    6. Kozubowski, Tomasz J. & Panorska, Anna K., 1996. "On moments and tail behavior of v-stable random variables," Statistics & Probability Letters, Elsevier, vol. 29(4), pages 307-315, September.
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