Pub Date : 2026-01-24DOI: 10.1016/j.spa.2026.104894
Martina Favero , Jere Koskela
The coalescent is a foundational model of latent genealogical trees under neutral evolution, but suffers from intractable sampling probabilities. Methods for approximating these sampling probabilities either introduce bias or fail to scale to large sample sizes. We show that a class of cost functionals of the coalescent with recurrent mutation and a finite number of alleles converge to tractable processes in the infinite-sample limit. A particular choice of costs yields insight about importance sampling methods, which are a classical tool for coalescent sampling probability approximation. These insights reveal that the behaviour of coalescent importance sampling algorithms differs markedly from standard sequential importance samplers, with or without resampling. We conduct a simulation study to verify that our asymptotics are accurate for algorithms with finite (and moderate) sample sizes. Our results constitute the first theoretical description of large-sample importance sampling algorithms for the coalescent, provide heuristics for the a priori optimisation of computational effort, and identify settings where resampling is harmful for algorithm performance. We observe strikingly different behaviour for importance sampling methods under the infinite sites model of mutation, which is regarded as a good and more tractable approximation of finite alleles mutation in most respects.
{"title":"Large-sample analysis of cost functionals for inference under the coalescent","authors":"Martina Favero , Jere Koskela","doi":"10.1016/j.spa.2026.104894","DOIUrl":"10.1016/j.spa.2026.104894","url":null,"abstract":"<div><div>The coalescent is a foundational model of latent genealogical trees under neutral evolution, but suffers from intractable sampling probabilities. Methods for approximating these sampling probabilities either introduce bias or fail to scale to large sample sizes. We show that a class of cost functionals of the coalescent with recurrent mutation and a finite number of alleles converge to tractable processes in the infinite-sample limit. A particular choice of costs yields insight about importance sampling methods, which are a classical tool for coalescent sampling probability approximation. These insights reveal that the behaviour of coalescent importance sampling algorithms differs markedly from standard sequential importance samplers, with or without resampling. We conduct a simulation study to verify that our asymptotics are accurate for algorithms with finite (and moderate) sample sizes. Our results constitute the first theoretical description of large-sample importance sampling algorithms for the coalescent, provide heuristics for the a priori optimisation of computational effort, and identify settings where resampling is harmful for algorithm performance. We observe strikingly different behaviour for importance sampling methods under the infinite sites model of mutation, which is regarded as a good and more tractable approximation of finite alleles mutation in most respects.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"195 ","pages":"Article 104894"},"PeriodicalIF":1.2,"publicationDate":"2026-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146078392","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-22DOI: 10.1016/j.spa.2026.104891
Petr Čoupek, Pavel Kříž, Matěj Svoboda
In this paper, a divergence-type integral of a random integrand with respect to the Hermite process of order with Hurst parameter H ∈ (1/2, 1) is defined and it is shown that the integral is of finite 1/H-variation.
{"title":"On the 1/H-variation of the divergence integral with respect to a Hermite process","authors":"Petr Čoupek, Pavel Kříž, Matěj Svoboda","doi":"10.1016/j.spa.2026.104891","DOIUrl":"10.1016/j.spa.2026.104891","url":null,"abstract":"<div><div>In this paper, a divergence-type integral of a random integrand with respect to the Hermite process of order <span><math><mrow><mi>k</mi><mo>∈</mo><mi>N</mi></mrow></math></span> with Hurst parameter <em>H</em> ∈ (1/2, 1) is defined and it is shown that the integral is of finite 1/<em>H</em>-variation.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"195 ","pages":"Article 104891"},"PeriodicalIF":1.2,"publicationDate":"2026-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146078432","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-22DOI: 10.1016/j.spa.2026.104892
Martin Friesen , Stefan Gerhold , Kristof Wiedermann
We study small-time central limit theorems for stochastic Volterra integral equations with Hölder continuous coefficients and general locally square integrable Volterra kernels. We prove the convergence of the finite-dimensional distributions, a functional CLT, and limit theorems for smooth transformations of the process, covering a large class of Volterra kernels including rough models based on Riemann-Liouville kernels with short- or long-range dependencies. To illustrate our results, we derive asymptotic pricing formulae for digital calls on the realized variance in three different regimes. The latter provides a robust and largely model-independent pricing method for small maturities in rough volatility models. Finally, for the case of completely monotone kernels, we introduce a flexible framework of Hilbert space-valued Markovian lifts and derive analogous limit theorems for such lifts. The latter provides new small-time limit theorems for stochastic Volterra processes obtained by transformation of the underlying Volterra kernels.
{"title":"Small-time central limit theorems for stochastic Volterra integral equations and their Markovian lifts","authors":"Martin Friesen , Stefan Gerhold , Kristof Wiedermann","doi":"10.1016/j.spa.2026.104892","DOIUrl":"10.1016/j.spa.2026.104892","url":null,"abstract":"<div><div>We study small-time central limit theorems for stochastic Volterra integral equations with Hölder continuous coefficients and general locally square integrable Volterra kernels. We prove the convergence of the finite-dimensional distributions, a functional CLT, and limit theorems for smooth transformations of the process, covering a large class of Volterra kernels including rough models based on Riemann-Liouville kernels with short- or long-range dependencies. To illustrate our results, we derive asymptotic pricing formulae for digital calls on the realized variance in three different regimes. The latter provides a robust and largely model-independent pricing method for small maturities in rough volatility models. Finally, for the case of completely monotone kernels, we introduce a flexible framework of Hilbert space-valued Markovian lifts and derive analogous limit theorems for such lifts. The latter provides new small-time limit theorems for stochastic Volterra processes obtained by transformation of the underlying Volterra kernels.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"195 ","pages":"Article 104892"},"PeriodicalIF":1.2,"publicationDate":"2026-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146078391","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-17DOI: 10.1016/j.spa.2026.104888
Zhishui Hu , Hanying Liang , Qiying Wang
On the convergence to stochastic integrals, semi-martingale structure is imposed in most of previous literature. This semi-martingale structure is restrictive in many statistical and econometric applications, particularly in the field of cointegration. In this paper, we investigate the convergence to stochastic integrals beyond the semi-martingale structure. In particular, we consider the convergence of stochastic integrals with general linear process innovations, allowing for long memory, short memory and antipersistence processes in a unified framework.
{"title":"Limit theorems for stochastic integrals with long memory processes","authors":"Zhishui Hu , Hanying Liang , Qiying Wang","doi":"10.1016/j.spa.2026.104888","DOIUrl":"10.1016/j.spa.2026.104888","url":null,"abstract":"<div><div>On the convergence to stochastic integrals, semi-martingale structure is imposed in most of previous literature. This semi-martingale structure is restrictive in many statistical and econometric applications, particularly in the field of cointegration. In this paper, we investigate the convergence to stochastic integrals beyond the semi-martingale structure. In particular, we consider the convergence of stochastic integrals with general linear process innovations, allowing for long memory, short memory and antipersistence processes in a unified framework.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"195 ","pages":"Article 104888"},"PeriodicalIF":1.2,"publicationDate":"2026-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146023275","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-16DOI: 10.1016/j.spa.2026.104889
Federico Girotti, Alessandro Vitale
In this work we exhibit a class of examples that show that the characterization of purification of quantum trajectories in terms of ‘dark’ subspaces that was proved for finite dimensional systems (Infin. Dimens. Anal. Quantum Probab. Relat. Top., 06(02), 223-243, 2003 and IMS Lectures Notes-Monograph Series, 48, 252-261, 2006) fails to hold in infinite dimensional ones. Moreover, we prove that the new phenomenon emerging in our class of models and preventing purification to happen is the only new possibility that appears in infinite dimensional systems. Our proof strategy points out that the presence of new phenomena in infinite dimensional systems is due to the fact that the set of orthogonal projections is not sequentially compact. Having in mind this insight, we are able to prove that the finite dimensional result extends to a class of infinite dimensional models.
{"title":"Purification of quantum trajectories in infinite dimensions","authors":"Federico Girotti, Alessandro Vitale","doi":"10.1016/j.spa.2026.104889","DOIUrl":"10.1016/j.spa.2026.104889","url":null,"abstract":"<div><div>In this work we exhibit a class of examples that show that the characterization of purification of quantum trajectories in terms of ‘dark’ subspaces that was proved for finite dimensional systems (<em>Infin. Dimens. Anal. Quantum Probab. Relat. Top.</em>, 06(02), 223-243, 2003 and <em>IMS Lectures Notes-Monograph Series</em>, 48, 252-261, 2006) fails to hold in infinite dimensional ones. Moreover, we prove that the new phenomenon emerging in our class of models and preventing purification to happen is the only new possibility that appears in infinite dimensional systems. Our proof strategy points out that the presence of new phenomena in infinite dimensional systems is due to the fact that the set of orthogonal projections is not sequentially compact. Having in mind this insight, we are able to prove that the finite dimensional result extends to a class of infinite dimensional models.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"195 ","pages":"Article 104889"},"PeriodicalIF":1.2,"publicationDate":"2026-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146023273","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-14DOI: 10.1016/j.spa.2026.104885
Merle Munko , Dennis Dobler
The functional delta-method has a wide range of applications in statistics. Applications on functionals of empirical processes yield various limit results for classical statistics. To improve the finite sample properties of statistical inference procedures that are based on the limit results, resampling procedures such as random permutation and bootstrap methods are a popular solution. In order to analyze the behaviour of the functionals of the resampling empirical processes, corresponding conditional functional delta-methods are desirable. While conditional functional delta-methods for some special cases already exist, there is a lack of more general conditional functional delta-methods for resampling procedures as the permutation and pooled bootstrap method. This gap is addressed in the present paper. Thereby, a general multiple sample problem is considered. The flexible application of the developed conditional delta-method is shown in various relevant examples.
{"title":"Conditional delta-method for resampling empirical processes in multiple sample problems","authors":"Merle Munko , Dennis Dobler","doi":"10.1016/j.spa.2026.104885","DOIUrl":"10.1016/j.spa.2026.104885","url":null,"abstract":"<div><div>The functional delta-method has a wide range of applications in statistics. Applications on functionals of empirical processes yield various limit results for classical statistics. To improve the finite sample properties of statistical inference procedures that are based on the limit results, resampling procedures such as random permutation and bootstrap methods are a popular solution. In order to analyze the behaviour of the functionals of the resampling empirical processes, corresponding conditional functional delta-methods are desirable. While conditional functional delta-methods for some special cases already exist, there is a lack of more general conditional functional delta-methods for resampling procedures as the permutation and pooled bootstrap method. This gap is addressed in the present paper. Thereby, a general multiple sample problem is considered. The flexible application of the developed conditional delta-method is shown in various relevant examples.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"195 ","pages":"Article 104885"},"PeriodicalIF":1.2,"publicationDate":"2026-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146023276","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-14DOI: 10.1016/j.spa.2026.104887
A. Pedicone, E. Orsingher
In this paper we study the telegraph meander, a random function obtained by conditioning the telegraph process to stay above the zero level. The finite dimensional distribution of the telegraph meander is derived by applying the reflection principle for the telegraph process and the Markovianity of the telegraph process with the velocity process. We show that the law of the telegraph meander at the end time is a solution to a hyperbolic equation, and we find the characteristic function and moments of any order. Finally, we prove that Brownian meander is the weak limit of the telegraph meander.
{"title":"On the distribution of the telegraph meander and its properties","authors":"A. Pedicone, E. Orsingher","doi":"10.1016/j.spa.2026.104887","DOIUrl":"10.1016/j.spa.2026.104887","url":null,"abstract":"<div><div>In this paper we study the telegraph meander, a random function obtained by conditioning the telegraph process to stay above the zero level. The finite dimensional distribution of the telegraph meander is derived by applying the reflection principle for the telegraph process and the Markovianity of the telegraph process with the velocity process. We show that the law of the telegraph meander at the end time is a solution to a hyperbolic equation, and we find the characteristic function and moments of any order. Finally, we prove that Brownian meander is the weak limit of the telegraph meander.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"195 ","pages":"Article 104887"},"PeriodicalIF":1.2,"publicationDate":"2026-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145978768","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-14DOI: 10.1016/j.spa.2026.104884
Zhang Chen , Bixiang Wang , Dandan Yang
This paper is concerned with the limiting behavior of the fractional stochastic reaction-diffusion equations defined in a sequence of open balls of radius k in . Under certain conditions, we prove that every weak limit point of invariant measures of the equations defined in Ok must be an invariant measure of the equation defined on as k → ∞. We also prove the convergence of invariant measures of the equations in Ok in terms of the Wasserstein metric and derive the rate of such convergence as k → ∞. The uniform tail-ends estimates of solutions are employed to overcome the non-compactness of Sobolev embeddings on .
{"title":"Limiting behavior of invariant measures of fractional stochastic reaction-diffusion equations on expanding domains","authors":"Zhang Chen , Bixiang Wang , Dandan Yang","doi":"10.1016/j.spa.2026.104884","DOIUrl":"10.1016/j.spa.2026.104884","url":null,"abstract":"<div><div>This paper is concerned with the limiting behavior of the fractional stochastic reaction-diffusion equations defined in a sequence <span><math><msubsup><mrow><mo>{</mo><msub><mi>O</mi><mi>k</mi></msub><mo>}</mo></mrow><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>∞</mi></msubsup></math></span> of open balls of radius <em>k</em> in <span><math><msup><mi>R</mi><mi>n</mi></msup></math></span>. Under certain conditions, we prove that every weak limit point of invariant measures of the equations defined in <em>O<sub>k</sub></em> must be an invariant measure of the equation defined on <span><math><msup><mi>R</mi><mi>n</mi></msup></math></span> as <em>k</em> → ∞. We also prove the convergence of invariant measures of the equations in <em>O<sub>k</sub></em> in terms of the Wasserstein metric and derive the rate of such convergence as <em>k</em> → ∞. The uniform tail-ends estimates of solutions are employed to overcome the non-compactness of Sobolev embeddings on <span><math><msup><mi>R</mi><mi>n</mi></msup></math></span>.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"195 ","pages":"Article 104884"},"PeriodicalIF":1.2,"publicationDate":"2026-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146023260","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-12DOI: 10.1016/j.spa.2026.104883
David Hobson , Dominykas Norgilas
<div><div>We give an injective martingale coupling; in particular, given measures <em>μ</em> and <em>ν</em> in convex order on <span><math><mi>R</mi></math></span> such that <em>ν</em> is continuous, we construct a martingale transport such that for each <em>y</em> in the support of the target law <em>ν</em> there is a <em>unique x</em> in a support of the initial law <em>μ</em> such that (some of) the mass at <em>x</em> is transported to <em>y</em>. Then <em>π</em> has disintegration <span><math><mrow><mi>π</mi><mrow><mo>(</mo><mi>d</mi><mi>x</mi><mo>,</mo><mi>d</mi><mi>y</mi><mo>)</mo></mrow><mo>=</mo><mi>ν</mi><mrow><mo>(</mo><mi>d</mi><mi>y</mi><mo>)</mo></mrow><msub><mi>δ</mi><mrow><mi>θ</mi><mo>(</mo><mi>y</mi><mo>)</mo></mrow></msub><mrow><mo>(</mo><mi>d</mi><mi>x</mi><mo>)</mo></mrow></mrow></math></span> for some function <em>θ</em>.</div><div>More precisely we construct a martingale coupling <em>π</em> of the measures <em>μ</em> and <em>ν</em> such that there is a set Γ<sub><em>μ</em></sub> such that <span><math><mrow><mi>μ</mi><mo>(</mo><msub><mstyle><mi>Γ</mi></mstyle><mi>μ</mi></msub><mo>)</mo><mo>=</mo><mn>1</mn></mrow></math></span> and a disintegration <span><math><msub><mrow><mo>(</mo><msub><mi>π</mi><mi>x</mi></msub><mo>)</mo></mrow><mrow><mi>x</mi><mo>∈</mo><msub><mstyle><mi>Γ</mi></mstyle><mi>μ</mi></msub></mrow></msub></math></span> of <em>π</em> of the form <span><math><mrow><mi>π</mi><mrow><mo>(</mo><mi>d</mi><mi>x</mi><mo>,</mo><mi>d</mi><mi>y</mi><mo>)</mo></mrow><mo>=</mo><msub><mi>π</mi><mi>x</mi></msub><mrow><mo>(</mo><mi>d</mi><mi>y</mi><mo>)</mo></mrow><mi>μ</mi><mrow><mo>(</mo><mi>d</mi><mi>x</mi><mo>)</mo></mrow></mrow></math></span> such that, with <span><math><msub><mstyle><mi>Γ</mi></mstyle><msub><mi>π</mi><mi>x</mi></msub></msub></math></span> a support of <em>π<sub>x</sub></em>, we have <span><math><mrow><mo>#</mo><mrow><mo>{</mo><mi>x</mi><mo>∈</mo><msub><mstyle><mi>Γ</mi></mstyle><mi>μ</mi></msub><mo>:</mo><mi>y</mi><mo>∈</mo><msub><mstyle><mi>Γ</mi></mstyle><msub><mi>π</mi><mi>x</mi></msub></msub><mo>}</mo></mrow><mo>∈</mo><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow></mrow></math></span> for all <em>y</em> and <span><math><mrow><mrow><mo>{</mo><mi>y</mi><mo>:</mo><mo>#</mo><mrow><mo>{</mo><mi>x</mi><mo>∈</mo><msub><mstyle><mi>Γ</mi></mstyle><mi>μ</mi></msub><mo>:</mo><mi>y</mi><mo>∈</mo><msub><mstyle><mi>Γ</mi></mstyle><msub><mi>π</mi><mi>x</mi></msub></msub><mo>}</mo></mrow><mo>=</mo><mn>1</mn><mo>}</mo></mrow><mo>=</mo><mtext>supp</mtext><mrow><mo>(</mo><mi>ν</mi><mo>)</mo></mrow></mrow></math></span>. Moreover, if <em>μ</em> is continuous we may take <span><math><mrow><msub><mstyle><mi>Γ</mi></mstyle><msub><mi>π</mi><mi>x</mi></msub></msub><mo>=</mo><mtext>supp</mtext><mrow><mo>(</mo><msub><mi>π</mi><mi>x</mi></msub><mo>)</mo></mrow></mrow></math></span> for each <em>x</em>. However, we cannot also insist that <span><math><mrow><msub><mstyle><mi>Γ</mi></mstyle><mi>μ</mi></msub><mo>=</m
{"title":"An injective martingale coupling","authors":"David Hobson , Dominykas Norgilas","doi":"10.1016/j.spa.2026.104883","DOIUrl":"10.1016/j.spa.2026.104883","url":null,"abstract":"<div><div>We give an injective martingale coupling; in particular, given measures <em>μ</em> and <em>ν</em> in convex order on <span><math><mi>R</mi></math></span> such that <em>ν</em> is continuous, we construct a martingale transport such that for each <em>y</em> in the support of the target law <em>ν</em> there is a <em>unique x</em> in a support of the initial law <em>μ</em> such that (some of) the mass at <em>x</em> is transported to <em>y</em>. Then <em>π</em> has disintegration <span><math><mrow><mi>π</mi><mrow><mo>(</mo><mi>d</mi><mi>x</mi><mo>,</mo><mi>d</mi><mi>y</mi><mo>)</mo></mrow><mo>=</mo><mi>ν</mi><mrow><mo>(</mo><mi>d</mi><mi>y</mi><mo>)</mo></mrow><msub><mi>δ</mi><mrow><mi>θ</mi><mo>(</mo><mi>y</mi><mo>)</mo></mrow></msub><mrow><mo>(</mo><mi>d</mi><mi>x</mi><mo>)</mo></mrow></mrow></math></span> for some function <em>θ</em>.</div><div>More precisely we construct a martingale coupling <em>π</em> of the measures <em>μ</em> and <em>ν</em> such that there is a set Γ<sub><em>μ</em></sub> such that <span><math><mrow><mi>μ</mi><mo>(</mo><msub><mstyle><mi>Γ</mi></mstyle><mi>μ</mi></msub><mo>)</mo><mo>=</mo><mn>1</mn></mrow></math></span> and a disintegration <span><math><msub><mrow><mo>(</mo><msub><mi>π</mi><mi>x</mi></msub><mo>)</mo></mrow><mrow><mi>x</mi><mo>∈</mo><msub><mstyle><mi>Γ</mi></mstyle><mi>μ</mi></msub></mrow></msub></math></span> of <em>π</em> of the form <span><math><mrow><mi>π</mi><mrow><mo>(</mo><mi>d</mi><mi>x</mi><mo>,</mo><mi>d</mi><mi>y</mi><mo>)</mo></mrow><mo>=</mo><msub><mi>π</mi><mi>x</mi></msub><mrow><mo>(</mo><mi>d</mi><mi>y</mi><mo>)</mo></mrow><mi>μ</mi><mrow><mo>(</mo><mi>d</mi><mi>x</mi><mo>)</mo></mrow></mrow></math></span> such that, with <span><math><msub><mstyle><mi>Γ</mi></mstyle><msub><mi>π</mi><mi>x</mi></msub></msub></math></span> a support of <em>π<sub>x</sub></em>, we have <span><math><mrow><mo>#</mo><mrow><mo>{</mo><mi>x</mi><mo>∈</mo><msub><mstyle><mi>Γ</mi></mstyle><mi>μ</mi></msub><mo>:</mo><mi>y</mi><mo>∈</mo><msub><mstyle><mi>Γ</mi></mstyle><msub><mi>π</mi><mi>x</mi></msub></msub><mo>}</mo></mrow><mo>∈</mo><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow></mrow></math></span> for all <em>y</em> and <span><math><mrow><mrow><mo>{</mo><mi>y</mi><mo>:</mo><mo>#</mo><mrow><mo>{</mo><mi>x</mi><mo>∈</mo><msub><mstyle><mi>Γ</mi></mstyle><mi>μ</mi></msub><mo>:</mo><mi>y</mi><mo>∈</mo><msub><mstyle><mi>Γ</mi></mstyle><msub><mi>π</mi><mi>x</mi></msub></msub><mo>}</mo></mrow><mo>=</mo><mn>1</mn><mo>}</mo></mrow><mo>=</mo><mtext>supp</mtext><mrow><mo>(</mo><mi>ν</mi><mo>)</mo></mrow></mrow></math></span>. Moreover, if <em>μ</em> is continuous we may take <span><math><mrow><msub><mstyle><mi>Γ</mi></mstyle><msub><mi>π</mi><mi>x</mi></msub></msub><mo>=</mo><mtext>supp</mtext><mrow><mo>(</mo><msub><mi>π</mi><mi>x</mi></msub><mo>)</mo></mrow></mrow></math></span> for each <em>x</em>. However, we cannot also insist that <span><math><mrow><msub><mstyle><mi>Γ</mi></mstyle><mi>μ</mi></msub><mo>=</m","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"195 ","pages":"Article 104883"},"PeriodicalIF":1.2,"publicationDate":"2026-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145978770","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-11DOI: 10.1016/j.spa.2026.104882
Mathew D. Penrose , Xiaochuan Yang
Let k, d be positive integers. We determine a sequence of constants that are asymptotic to the probability that the cluster at the origin in a d-dimensional Poisson Boolean model with balls of fixed radius is of order k, as the intensity becomes large. Using this, we determine the asymptotics of the mean of the number of components of order k, denoted Sn,k in a random geometric graph on n uniformly distributed vertices in a smoothly bounded compact region of d-dimensional Euclidean space, with distance parameter r(n) chosen so that the expected degree grows slowly as n becomes large (the so-called mildly dense limiting regime). We also show that the variance of Sn,k is asymptotic to its mean, and prove Poisson and normal approximation results for Sn,k in this limiting regime. We provide analogous results for the corresponding Poisson process (i.e. with a Poisson number of points).
We also give similar results in the so-called mildly sparse limiting regime where r(n) is chosen so the expected degree decays slowly to zero as n becomes large.
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