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R₊ is the weak limit of a sequence of distributions with rational Laplace-Stieltjes transform (of mixtures with positive weights of gamma distribution). Steutel and van Eenige [Comm. Stat.-Stochastic Models 13 (1997), 271-274] looked at the discrete analogue of this result. Here we look into the role of the discrete Mittag-Leffler distributions in the study of approximation of distributions on Z₊. The concept of Poisson mixtures is generalized to discrete stable mixtures. It is shown that distributions on Z₊ that can be approximated by mixtures of generalized discrete Linnik distributions are discrete stable mixtures. Discrete semi-stable autoregressive process is introduced. As a special case, we study discrete stable autoregressive process.

Advances and Applications in Statistics
Volume 3, Issue 3, Pages 255 - 266 (December 2003)

ON DISCRETE STABLE LAWS
K. Jayakumar (India) and P. C. Sreenivas (India)

Abstract:
It is known that every distribution on R₊ is the weak limit of a sequence of distributions with rational Laplace-Stieltjes transform (of mixtures with positive weights of gamma distribution). Steutel and van Eenige [Comm. Stat.-Stochastic Models 13 (1997), 271-274] looked at the discrete analogue of this result. Here we look into the role of the discrete Mittag-Leffler distributions in the study of approximation of distributions on Z₊. The concept of Poisson mixtures is generalized to discrete stable mixtures. It is shown that distributions on Z₊ that can be approximated by mixtures of generalized discrete Linnik distributions are discrete stable mixtures. Discrete semi-stable autoregressive process is introduced. As a special case, we study discrete stable autoregressive process.

Keywords and phrases: autoregressive models, discrete Mittag-Leffler laws, discrete semi- stable laws, discrete stable laws, positive Linnik distribution, Mittag-Leffler distribution, Poisson mixtures.

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