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On First Order Autoregressive Asymmetric Logistic Process. JOURNAL OF THE INDIAN SOCIETY FOR PROBABILITY AND STATISTICS 2023. [DOI: 10.1007/s41096-023-00147-8] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 02/05/2023]
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Single-Block Recursive Poisson–Dirichlet Fragmentations of Normalized Generalized Gamma Processes. MATHEMATICS 2022. [DOI: 10.3390/math10040561] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 02/04/2023]
Abstract
Dong, Goldschmidt and Martin (2006) (DGM) showed that, for 0<α<1, and θ>−α, the repeated application of independent single-block fragmentation operators based on mass partitions following a two-parameter Poisson–Dirichlet distribution with parameters (α,1−α) to a mass partition having a Poisson–Dirichlet distribution with parameters (α,θ) leads to a remarkable nested family of Poisson—Dirichlet distributed mass partitions with parameters (α,θ+r) for r=0,1,2,⋯. Furthermore, these generate a Markovian sequence of α-diversities following Mittag-Leffler distributions, whose ratios lead to independent Beta-distributed variables. These Markov chains are referred to as Mittag-Leffler Markov chains and arise in the broader literature involving Pólya urn and random tree/graph growth models. Here we obtain explicit descriptions of properties of these processes when conditioned on a mixed Poisson process when it equates to an integer n, which has interpretations in a species sampling context. This is equivalent to obtaining properties of the fragmentation operations of (DGM) when applied to mass partitions formed by the normalized jumps of a generalized gamma subordinator and its generalizations. We focus primarily on the case where n=0,1.
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Gibbs partitions, Riemann–Liouville fractional operators, Mittag–Leffler functions, and fragmentations derived from stable subordinators. J Appl Probab 2021. [DOI: 10.1017/jpr.2020.93] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
Abstract
AbstractPitman (2003), and subsequently Gnedin and Pitman (2006), showed that a large class of random partitions of the integers derived from a stable subordinator of index $\alpha\in(0,1)$ have infinite Gibbs (product) structure as a characterizing feature. The most notable case are random partitions derived from the two-parameter Poisson–Dirichlet distribution, $\textrm{PD}(\alpha,\theta)$, whose corresponding $\alpha$-diversity/local time have generalized Mittag–Leffler distributions, denoted by $\textrm{ML}(\alpha,\theta)$. Our aim in this work is to provide indications on the utility of the wider class of Gibbs partitions as it relates to a study of Riemann–Liouville fractional integrals and size-biased sampling, and in decompositions of special functions, and its potential use in the understanding of various constructions of more exotic processes. We provide characterizations of general laws associated with nested families of $\textrm{PD}(\alpha,\theta)$ mass partitions that are constructed from fragmentation operations described in Dong et al. (2014). These operations are known to be related in distribution to various constructions of discrete random trees/graphs in [n], and their scaling limits. A centerpiece of our work is results related to Mittag–Leffler functions, which play a key role in fractional calculus and are otherwise Laplace transforms of the $\textrm{ML}(\alpha,\theta)$ variables. Notably, this leads to an interpretation within the context of $\textrm{PD}(\alpha,\theta)$ laws conditioned on Poisson point process counts over intervals of scaled lengths of the $\alpha$-diversity.
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Barabesi L. The computation of the probability density and distribution functions for some families of random variables by means of the Wynn-ρ accelerated Post-Widder formula. COMMUN STAT-SIMUL C 2018. [DOI: 10.1080/03610918.2018.1496254] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/27/2022]
Affiliation(s)
- Lucio Barabesi
- Department of Economics and Statistics, University of Siena, Piazza San Francesco, Siena, Italy
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Pitman J, Yakubovich Y. Extremes and gaps in sampling from a GEM random discrete distribution. ELECTRON J PROBAB 2017. [DOI: 10.1214/17-ejp59] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Barabesi L, Cerasa A, Cerioli A, Perrotta D. A new family of tempered distributions. Electron J Stat 2016. [DOI: 10.1214/16-ejs1214] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Favaro S, Lijoi A, Prünster I. Conditional formulae for Gibbs-type exchangeable random partitions. ANN APPL PROBAB 2013. [DOI: 10.1214/12-aap843] [Citation(s) in RCA: 19] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Haas B, Rivero V. Quasi-stationary distributions and Yaglom limits of self-similar Markov processes. Stoch Process Their Appl 2012. [DOI: 10.1016/j.spa.2012.08.006] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
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