<p>The Ewens-Pitman model is a probability distribution for random partitions of the set <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$[n]=\{1,\ldots ,n\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mo stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>, parameterized by <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\alpha \in [0,1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\theta >-\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>θ</mi> <mo>></mo> <mo>-</mo> <mi>α</mi> </mrow> </math></EquationSource> </InlineEquation>, with <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\alpha =0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> corresponding to the Ewens model in population genetics. The goal of this paper is to provide an alternative and concise proof of the Feng-Hoppe large deviation principle for the number <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$K_{n}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>K</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> of partition sets in the Ewens-Pitman model with <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\alpha \in (0,1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\theta >-\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>θ</mi> <mo>></mo> <mo>-</mo> <mi>α</mi> </mrow> </math></EquationSource> </InlineEquation>. Our approach leverages an integral representation of the moment-generating function of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$K_{n}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>K</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> in terms of the (one-parameter) Mittag-Leffler function, along with a sharp asymptotic expansion of it. This approach significantly simplifies the original proof of Feng-Hoppe large deviation principle, as it avoids all the technical difficulties arising from a continuity argument with respect to rational and non-rational values of the parameter <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>. Beyond large deviations for <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$K_{n}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>K</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation>, our approach allows to establish a sharp concentration inequality for <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$K_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>K</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> involving the rate function of the large deviation principle.</p>

A new look on large deviations and concentration inequalities for Ewens-Pitman model

期刊论文
发表日期：

2026年7月9日

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Bernard Bercu,
Stefano Favaro

摘要

The Ewens-Pitman model is a probability distribution for random partitions of the set $[n]=\{1,\ldots ,n\}$ $[n] = {1, \dots, n}$ , parameterized by $\alpha \in [0,1)$ $α \in [0, 1)$ and $\theta >-\alpha $ $θ > - α$ , with $\alpha =0$ $α = 0$ corresponding to the Ewens model in population genetics. The goal of this paper is to provide an alternative and concise proof of the Feng-Hoppe large deviation principle for the number $K_{n}$ $K_{n}$ of partition sets in the Ewens-Pitman model with $\alpha \in (0,1)$ $α \in (0, 1)$ and $\theta >-\alpha $ $θ > - α$ . Our approach leverages an integral representation of the moment-generating function of $K_{n}$ $K_{n}$ in terms of the (one-parameter) Mittag-Leffler function, along with a sharp asymptotic expansion of it. This approach significantly simplifies the original proof of Feng-Hoppe large deviation principle, as it avoids all the technical difficulties arising from a continuity argument with respect to rational and non-rational values of the parameter $\alpha $ $α$ . Beyond large deviations for $K_{n}$ $K_{n}$ , our approach allows to establish a sharp concentration inequality for $K_n$ $K_{n}$ involving the rate function of the large deviation principle.