Asymptotic Properties of Random Restricted Partitions

Tiefeng Jiang; Ke Wang

doi:10.3390/math11020417

Asymptotic Properties of Random Restricted Partitions

Tiefeng Jiang, Ke Wang

Statistics (Twin Cities)

Research output: Contribution to journal › Article › peer-review

Abstract

We study two types of probability measures on the set of integer partitions of n with at most m parts. The first one chooses the partition with a chance related to its largest part only. We obtain the limiting distributions of all of the parts together and that of the largest part as n tending to infinity for m fixed or tending to infinity with (Formula presented.). In particular, if m goes to infinity not too fast, the largest part satisfies the central limit theorem. The second measure is very general and includes the Dirichlet and uniform distributions as special cases. The joint asymptotic distributions of the parts are derived by taking limits of n and m in the same manner as that in the first probability measure.

Original language	English (US)
Article number	417
Journal	Mathematics
Volume	11
Issue number	2
DOIs	https://doi.org/10.3390/math11020417
State	Published - Jan 2023

Bibliographical note

Funding Information:
This work was supported by the National Science Foundation (NSF) Grant [DMS-1916014 to T.J., DMS-1406279 to T.J., DMS-2210802 to T.J.]; and by the Research Grants Council (RGC) of Hong Kong [GRF 16308219 to K.W., GRF 16304222 to K.W., ECS 26304920 to K.W.].

Publisher Copyright:
© 2023 by the authors.

Keywords

asymptotic distributions
limit laws
random partitions

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abstract = "We study two types of probability measures on the set of integer partitions of n with at most m parts. The first one chooses the partition with a chance related to its largest part only. We obtain the limiting distributions of all of the parts together and that of the largest part as n tending to infinity for m fixed or tending to infinity with (Formula presented.). In particular, if m goes to infinity not too fast, the largest part satisfies the central limit theorem. The second measure is very general and includes the Dirichlet and uniform distributions as special cases. The joint asymptotic distributions of the parts are derived by taking limits of n and m in the same manner as that in the first probability measure.",

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N2 - We study two types of probability measures on the set of integer partitions of n with at most m parts. The first one chooses the partition with a chance related to its largest part only. We obtain the limiting distributions of all of the parts together and that of the largest part as n tending to infinity for m fixed or tending to infinity with (Formula presented.). In particular, if m goes to infinity not too fast, the largest part satisfies the central limit theorem. The second measure is very general and includes the Dirichlet and uniform distributions as special cases. The joint asymptotic distributions of the parts are derived by taking limits of n and m in the same manner as that in the first probability measure.

AB - We study two types of probability measures on the set of integer partitions of n with at most m parts. The first one chooses the partition with a chance related to its largest part only. We obtain the limiting distributions of all of the parts together and that of the largest part as n tending to infinity for m fixed or tending to infinity with (Formula presented.). In particular, if m goes to infinity not too fast, the largest part satisfies the central limit theorem. The second measure is very general and includes the Dirichlet and uniform distributions as special cases. The joint asymptotic distributions of the parts are derived by taking limits of n and m in the same manner as that in the first probability measure.

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Asymptotic Properties of Random Restricted Partitions

Abstract

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Keywords

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