Two-sided immigration, emigration and symmetry properties of self-similar interval partition evolutions
Forman et al. (2020+) constructed (α, θ)-interval partition evolutions for α ∈ (0, 1) and θ ≥ 0, in which the total sums of interval lengths (�total mass�) evolve as squared Bessel processes of dimension 2θ, where θ ≥ 0 acts as an immigration parameter. These evolutions have pseudostationary distrib...
| Main Authors: | , |
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| Format: | Journal article |
| Language: | English |
| Published: |
Instituto Nacional de Matemática Pura e Aplicada
2023
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| _version_ | 1797109815626956800 |
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| author | Shi, Q Winkel, M |
| author_facet | Shi, Q Winkel, M |
| author_sort | Shi, Q |
| collection | OXFORD |
| description | Forman et al. (2020+) constructed (α, θ)-interval partition evolutions for α ∈ (0, 1) and
θ ≥ 0, in which the total sums of interval lengths (�total mass�) evolve as squared Bessel processes
of dimension 2θ, where θ ≥ 0 acts as an immigration parameter. These evolutions have pseudostationary distributions related to regenerative Poisson�Dirichlet interval partitions. In this paper
we study symmetry properties of (α, θ)-interval partition evolutions. Furthermore, we introduce a
three-parameter family SSIP(α)
(θ1, θ2) of self-similar interval partition evolutions that have separate
left and right immigration parameters θ1 ≥ 0 and θ2 ≥ 0. They also have squared Bessel total
mass processes of dimension 2θ, where θ = θ1 + θ2 − α ≥ −α includes the usual parameter range
of the two-parameter Poisson�Dirichlet distribution � negative θ can be interpreted as an overall
emigration. Under the constraint max{θ1, θ2} ≥ α, we prove that an SSIP(α)
(θ1, θ2)-evolution is
pseudo-stationary for a new distribution on interval partitions, whose ranked sequence of lengths has
Poisson�Dirichlet distribution with parameters α and θ, but we are unable to cover all parameters
without developing a limit theory for composition-valued Markov chains, which we do in a sequel
paper. |
| first_indexed | 2024-03-07T07:46:35Z |
| format | Journal article |
| id | oxford-uuid:2e406c6c-683a-477c-b6aa-077130350783 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T07:46:35Z |
| publishDate | 2023 |
| publisher | Instituto Nacional de Matemática Pura e Aplicada |
| record_format | dspace |
| spelling | oxford-uuid:2e406c6c-683a-477c-b6aa-0771303507832023-06-19T12:03:29ZTwo-sided immigration, emigration and symmetry properties of self-similar interval partition evolutionsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:2e406c6c-683a-477c-b6aa-077130350783EnglishSymplectic ElementsInstituto Nacional de Matemática Pura e Aplicada2023Shi, QWinkel, MForman et al. (2020+) constructed (α, θ)-interval partition evolutions for α ∈ (0, 1) and θ ≥ 0, in which the total sums of interval lengths (�total mass�) evolve as squared Bessel processes of dimension 2θ, where θ ≥ 0 acts as an immigration parameter. These evolutions have pseudostationary distributions related to regenerative Poisson�Dirichlet interval partitions. In this paper we study symmetry properties of (α, θ)-interval partition evolutions. Furthermore, we introduce a three-parameter family SSIP(α) (θ1, θ2) of self-similar interval partition evolutions that have separate left and right immigration parameters θ1 ≥ 0 and θ2 ≥ 0. They also have squared Bessel total mass processes of dimension 2θ, where θ = θ1 + θ2 − α ≥ −α includes the usual parameter range of the two-parameter Poisson�Dirichlet distribution � negative θ can be interpreted as an overall emigration. Under the constraint max{θ1, θ2} ≥ α, we prove that an SSIP(α) (θ1, θ2)-evolution is pseudo-stationary for a new distribution on interval partitions, whose ranked sequence of lengths has Poisson�Dirichlet distribution with parameters α and θ, but we are unable to cover all parameters without developing a limit theory for composition-valued Markov chains, which we do in a sequel paper. |
| spellingShingle | Shi, Q Winkel, M Two-sided immigration, emigration and symmetry properties of self-similar interval partition evolutions |
| title | Two-sided immigration, emigration and symmetry properties of self-similar interval partition evolutions |
| title_full | Two-sided immigration, emigration and symmetry properties of self-similar interval partition evolutions |
| title_fullStr | Two-sided immigration, emigration and symmetry properties of self-similar interval partition evolutions |
| title_full_unstemmed | Two-sided immigration, emigration and symmetry properties of self-similar interval partition evolutions |
| title_short | Two-sided immigration, emigration and symmetry properties of self-similar interval partition evolutions |
| title_sort | two sided immigration emigration and symmetry properties of self similar interval partition evolutions |
| work_keys_str_mv | AT shiq twosidedimmigrationemigrationandsymmetrypropertiesofselfsimilarintervalpartitionevolutions AT winkelm twosidedimmigrationemigrationandsymmetrypropertiesofselfsimilarintervalpartitionevolutions |