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Transition density of an infinite-dimensional diffusion with the jack parameter

Part of: Combinatorial probability Markov processes

Published online by Cambridge University Press:  16 January 2023

Youzhou Zhou
Youzhou Zhou*
Affiliation:
Xi’an Jiaotong-Liverpool University
*
*Postal address: 111 Renai Road, Suzhou, Jiangsu, China 215 123. Email address: Youzhou.zhou@xjtlu.edu.cn
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Abstract

From the Poisson–Dirichlet diffusions to the Z-measure diffusions, they all have explicit transition densities. We show that the transition densities of the Z-measure diffusions can also be expressed as a mixture of a sequence of probability measures on the Thoma simplex. The coefficients are the same as the coefficients in the Poisson–Dirichlet diffusions. This fact will be uncovered by a dual process method in a special case where the Z-measure diffusions are established through an up–down chain in the Young graph.

Keywords

Jack graphtransition densitydual processup–down Markov chainKingman coalescent

MSC classification

Primary: 60J60: Diffusion processes
Secondary: 60C05: Combinatorial probability

Information

Type
Original Article
Information
Journal of Applied Probability , Volume 60 , Issue 3 , September 2023 , pp. 797 - 811
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Blei, D. M., Ng, A. Y. and Jordan, M. I. (2003). Latent Dirichlet allocation. J. Mach. Learn. Res. 3, 993–1022.Google Scholar
Borodin, A. and Olshanski, G. (2005). Z-measures on partitions and their scaling limits. Europ. J. Combin. 26, 795–834.CrossRefGoogle Scholar
Borodin, A. and Olshanski, G. (2009). Infinite-dimensional diffusions as limits of random walks on partitions. Prob. Theory Relat. Fields 144, 281–318.CrossRefGoogle Scholar
Ethier, S. N. (1992). Eigenstructure of the infinitely-many-neutral-alleles diffusion model. J. Appl. Prob. 29, 487–498.CrossRefGoogle Scholar
Ethier, S. N. and Kurtz, T. G. (1986). Markov Process: Characterization and Convergence. Wiley, New York.CrossRefGoogle Scholar
Feng, S., Sun, W., Wang, F. Y. and Xu, F. (2011). Functional inequalities for the two-parameter extension of the infinitely-many-neutral-alleles diffusion. J. Funct. Anal. 260, 39–413.CrossRefGoogle Scholar
Griffiths, R., Spanó, D., Ruggiero, M. and Zhou, Y. Dual process in the two-parameter Poisson–Dirichlet diffusion. Preprint, arXiv:2102.08520.Google Scholar
Kerov, S., Okounkov, A. and Olshanski, G. (1988). The boundary of Young graph with Jack edge multiplicities. Int. Math. Res. Not. 1998, 173–199.CrossRefGoogle Scholar
Kingman, J. F. C. (1975). The random discrete distributions. J. R. Statist. Soc. B 37, 1–22.Google Scholar
Kingman, J. F. C. (1978). The representation of partition structures. J. London Math. Soc. 18, 374–380.CrossRefGoogle Scholar
Korotkikh, S. Y. (2020). Transition functions of diffusion processes on the Thoma simplex. Funct. Anal. Appl. 54, 118–134.CrossRefGoogle Scholar
MacDonald, I. (1995). Symmetric Functions and Hall Polynomials. Cambridge University Press.Google Scholar
Olshanski, G. (2009). Anisotropic Young diagrams and infinite-dimensional diffusion processes with the Jack parameter. Int. Math. Res. Not. 2010, 1102–1166.Google Scholar
Petrov, L. (2009). A two-parameter family of infinite-dimensional diffusions in the Kingman simplex. Funct. Anal. Appl. 43, 45–66.CrossRefGoogle Scholar
Pitman, J. (1995). Exchangeable and partially exchangeable random partitions. Prob. Theory Relat. Fields 102, 145–158.CrossRefGoogle Scholar
Tavaré, S. (1984). Line-of-descent and genealogical processes, and their application in population genetics models. Theoret. Pop. Biol. 26, 119–164.CrossRefGoogle ScholarPubMed
Zhou, Y. (2015). Ergodic inequality of a two-parameter infinitely-many-alleles diffusion model. J. Appl. Prob. 52, 238–246.CrossRefGoogle Scholar