Pub Date : 2025-11-07DOI: 10.1016/j.spa.2025.104828
Wen Sun
We study the condensation phenomenon for the invariant measures of the mean-field model of reversible coagulation–fragmentation processes conditioned to a supercritical density of particles. It is shown that when the parameters of the associated balance equation satisfy a subexponential tail condition, there is a single giant particle that corresponds to the missing mass in the macroscopic limit. We also show that in this case, the rest of the particles are asymptotically i.i.d according to the normalised equilibrium state of the limit hydrodynamic differential equation. Conditions for the normal fluctuations and the -stable fluctuations around the condensed mass are given. We obtain the large deviation principle for the empirical measure of the masses of the particles at equilibrium as well.
{"title":"On the Condensation and fluctuations in reversible coagulation–fragmentation models","authors":"Wen Sun","doi":"10.1016/j.spa.2025.104828","DOIUrl":"10.1016/j.spa.2025.104828","url":null,"abstract":"<div><div>We study the condensation phenomenon for the invariant measures of the mean-field model of reversible coagulation–fragmentation processes conditioned to a supercritical density of particles. It is shown that when the parameters of the associated balance equation satisfy a subexponential tail condition, there is a single giant particle that corresponds to the missing mass in the macroscopic limit. We also show that in this case, the rest of the particles are asymptotically <em>i.i.d</em> according to the normalised equilibrium state of the limit hydrodynamic differential equation. Conditions for the normal fluctuations and the <span><math><mi>α</mi></math></span>-stable fluctuations around the condensed mass are given. We obtain the large deviation principle for the empirical measure of the masses of the particles at equilibrium as well.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"192 ","pages":"Article 104828"},"PeriodicalIF":1.2,"publicationDate":"2025-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145528793","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 : 2025-11-07DOI: 10.1016/j.spa.2025.104825
Qingwei Liu , Nicolas Privault
We derive normal approximation bounds for generalized -statistics of the form where and are independent sequences of i.i.d. random variables. Our approach relies on moment identities and cumulant bounds that are derived using partition diagram arguments. Normal approximation bounds in the Kolmogorov distance and moderate deviation results are then obtained by the cumulant method. Those results are applied to subgraph counting in the binomial random-connection model, which is a generalization of the Erdős–Rényi model.
{"title":"Gaussian fluctuations of generalized U-statistics and subgraph counting in the binomial random-connection model","authors":"Qingwei Liu , Nicolas Privault","doi":"10.1016/j.spa.2025.104825","DOIUrl":"10.1016/j.spa.2025.104825","url":null,"abstract":"<div><div>We derive normal approximation bounds for generalized <span><math><mi>U</mi></math></span>-statistics of the form <span><span><span><math><mrow><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi></mrow></msub><mrow><mo>(</mo><mi>f</mi><mo>)</mo></mrow><mo>≔</mo><munder><mrow><mo>∑</mo></mrow><mrow><mfrac><mrow><mn>1</mn><mo>≤</mo><mi>β</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow><mo>,</mo><mo>…</mo><mo>,</mo><mi>β</mi><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow><mo>≤</mo><mi>n</mi></mrow><mrow><mi>β</mi><mrow><mo>(</mo><mi>i</mi><mo>)</mo></mrow><mo>≠</mo><mi>β</mi><mrow><mo>(</mo><mi>j</mi><mo>)</mo></mrow><mo>,</mo><mspace></mspace><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≠</mo><mi>j</mi><mo>≤</mo><mi>k</mi></mrow></mfrac></mrow></munder><mi>f</mi><mrow><mo>(</mo><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>β</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>β</mi><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></mrow></msub><mo>,</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>β</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow><mo>,</mo><mi>β</mi><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>β</mi><mrow><mo>(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mo>,</mo><mi>β</mi><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></mrow></msub></mrow><mo>)</mo></mrow><mo>,</mo></mrow></math></span></span></span>where <span><math><msub><mrow><mrow><mo>{</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>}</mo></mrow></mrow><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>n</mi></mrow></msub></math></span> and <span><math><msub><mrow><mrow><mo>{</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mo>}</mo></mrow></mrow><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo><</mo><mi>j</mi><mo>≤</mo><mi>n</mi></mrow></msub></math></span> are independent sequences of i.i.d. random variables. Our approach relies on moment identities and cumulant bounds that are derived using partition diagram arguments. Normal approximation bounds in the Kolmogorov distance and moderate deviation results are then obtained by the cumulant method. Those results are applied to subgraph counting in the binomial random-connection model, which is a generalization of the Erdős–Rényi model.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"192 ","pages":"Article 104825"},"PeriodicalIF":1.2,"publicationDate":"2025-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145528823","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 : 2025-11-05DOI: 10.1016/j.spa.2025.104817
Lena Kuwata
In this work, we introduce a spatial branching process to model the growth of the mycelial network of a filamentous fungus. In this model, each filament is described by the position of its tip, the trajectory of which is solution to a stochastic differential equation with a drift term which depends on all the other trajectories. Each filament can branch either at its tip or along its length, that is to say at some past position of its tip, at some time- and space-dependent rates. It can stop growing at some rate which also depends on the positions of the other tips. We first construct the measure-valued process corresponding to this dynamics, then we study its large population limit and we characterise the limiting process as the weak solution to a system of partial differential equations.
{"title":"A generalised spatial branching process with ancestral branching to model the growth of a filamentous fungus","authors":"Lena Kuwata","doi":"10.1016/j.spa.2025.104817","DOIUrl":"10.1016/j.spa.2025.104817","url":null,"abstract":"<div><div>In this work, we introduce a spatial branching process to model the growth of the mycelial network of a filamentous fungus. In this model, each filament is described by the position of its tip, the trajectory of which is solution to a stochastic differential equation with a drift term which depends on all the other trajectories. Each filament can branch either at its tip or along its length, that is to say at some past position of its tip, at some time- and space-dependent rates. It can stop growing at some rate which also depends on the positions of the other tips. We first construct the measure-valued process corresponding to this dynamics, then we study its large population limit and we characterise the limiting process as the weak solution to a system of partial differential equations.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"192 ","pages":"Article 104817"},"PeriodicalIF":1.2,"publicationDate":"2025-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145528790","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 : 2025-11-05DOI: 10.1016/j.spa.2025.104815
Christian Hirsch , Takashi Owada
This paper considers limit theorems associated with subgraph counts in the age-dependent random connection model. First, we identify regimes where the count of sub-trees converges weakly to a stable random variable under suitable assumptions on the shape of trees. The proof relies on an intermediate result on weak convergence of associated point processes towards a Poisson point process. Additionally, we prove the same type of results for the clique counts. Here, a crucial ingredient includes the expectation asymptotics for clique counts, which itself is a result of independent interest.
{"title":"Limit theorems under heavy-tailed scenario in the age-dependent random connection models","authors":"Christian Hirsch , Takashi Owada","doi":"10.1016/j.spa.2025.104815","DOIUrl":"10.1016/j.spa.2025.104815","url":null,"abstract":"<div><div>This paper considers limit theorems associated with subgraph counts in the age-dependent random connection model. First, we identify regimes where the count of sub-trees converges weakly to a stable random variable under suitable assumptions on the shape of trees. The proof relies on an intermediate result on weak convergence of associated point processes towards a Poisson point process. Additionally, we prove the same type of results for the clique counts. Here, a crucial ingredient includes the expectation asymptotics for clique counts, which itself is a result of independent interest.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"192 ","pages":"Article 104815"},"PeriodicalIF":1.2,"publicationDate":"2025-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145528754","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 : 2025-10-30DOI: 10.1016/j.spa.2025.104814
Ranieri Dugo , Giacomo Giorgio , Paolo Pigato
Starting from the notion of multivariate fractional Brownian Motion introduced in [F. Lavancier, A. Philippe, and D. Surgailis. Covariance function of vector self-similar processes. Statistics & Probability Letters, 2009] we define a multivariate version of the fractional Ornstein–Uhlenbeck process. This multivariate Gaussian process is stationary, ergodic and allows for different Hurst exponents on each component. We characterize its correlation matrix and its short and long time asymptotics. Besides the marginal parameters, the cross correlation between one-dimensional marginal components is ruled by two parameters. We consider the problem of their inference, proposing two types of estimator, constructed from discrete observations of the process. We establish their asymptotic theory, in one case in the long time asymptotic setting, in the other case in the infill and long time asymptotic setting. The limit behavior can be asymptotically Gaussian or non-Gaussian, depending on the values of the Hurst exponents of the marginal components. The technical core of the paper relies on the analysis of asymptotic properties of functionals of Gaussian processes, that we establish using Malliavin calculus and Stein’s method. We provide numerical experiments that support our theoretical analysis and also suggest a conjecture on the application of one of these estimators to the multivariate fractional Brownian Motion.
从[F]中引入的多元分数布朗运动的概念出发。Lavancier， A. Philippe和D. Surgailis。向量自相似过程的协方差函数。统计&概率信件，2009]我们定义了分数Ornstein-Uhlenbeck过程的多元版本。这个多元高斯过程是平稳的，遍历的，并且允许在每个分量上有不同的Hurst指数。我们刻画了它的相关矩阵及其短、长时间渐近性。除了边缘参数外，一维边缘分量之间的相互关系还由两个参数决定。我们考虑了他们的推理问题，提出了两种类型的估计量，由过程的离散观测构造。我们建立了它们的渐近理论，一种是在长时间渐近设置下，另一种是在填充和长时间渐近设置下。极限行为可以是渐近高斯或非高斯的，这取决于边缘分量的Hurst指数的值。本文的技术核心是利用Malliavin演算和Stein方法建立高斯过程泛函的渐近性质分析。我们提供了数值实验来支持我们的理论分析，并提出了一个关于这些估计器在多元分数布朗运动中的应用的猜想。
{"title":"The multivariate fractional Ornstein–Uhlenbeck process","authors":"Ranieri Dugo , Giacomo Giorgio , Paolo Pigato","doi":"10.1016/j.spa.2025.104814","DOIUrl":"10.1016/j.spa.2025.104814","url":null,"abstract":"<div><div>Starting from the notion of multivariate fractional Brownian Motion introduced in [F. Lavancier, A. Philippe, and D. Surgailis. Covariance function of vector self-similar processes. Statistics & Probability Letters, 2009] we define a multivariate version of the fractional Ornstein–Uhlenbeck process. This multivariate Gaussian process is stationary, ergodic and allows for different Hurst exponents on each component. We characterize its correlation matrix and its short and long time asymptotics. Besides the marginal parameters, the cross correlation between one-dimensional marginal components is ruled by two parameters. We consider the problem of their inference, proposing two types of estimator, constructed from discrete observations of the process. We establish their asymptotic theory, in one case in the long time asymptotic setting, in the other case in the infill and long time asymptotic setting. The limit behavior can be asymptotically Gaussian or non-Gaussian, depending on the values of the Hurst exponents of the marginal components. The technical core of the paper relies on the analysis of asymptotic properties of functionals of Gaussian processes, that we establish using Malliavin calculus and Stein’s method. We provide numerical experiments that support our theoretical analysis and also suggest a conjecture on the application of one of these estimators to the multivariate fractional Brownian Motion.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"192 ","pages":"Article 104814"},"PeriodicalIF":1.2,"publicationDate":"2025-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145528791","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 : 2025-10-24DOI: 10.1016/j.spa.2025.104792
Ahmed El Alaoui
We show that in the Ising pure -spin model of spin glasses, shattering takes place at all inverse temperatures when is sufficiently large as a function of . Of special interest is the lower boundary of this interval which matches the large asymptotics of the inverse temperature marking the hypothetical dynamical transition predicted in statistical physics. We show this as a consequence of a ‘soft’ version of the overlap gap property which asserts the existence of a distance gap of points of typical energy from a typical sample from the Gibbs measure. We further show that this latter property implies that stable algorithms seeking to return a point of at least typical energy are confined to an exponentially rare subset of that super-level set, provided that their success probability is not vanishingly small.
{"title":"Near-optimal shattering in the Ising pure p-spin and rarity of solutions returned by stable algorithms","authors":"Ahmed El Alaoui","doi":"10.1016/j.spa.2025.104792","DOIUrl":"10.1016/j.spa.2025.104792","url":null,"abstract":"<div><div>We show that in the Ising pure <span><math><mi>p</mi></math></span>-spin model of spin glasses, shattering takes place at all inverse temperatures <span><math><mrow><mi>β</mi><mo>∈</mo><mrow><mo>(</mo><msqrt><mrow><mrow><mo>(</mo><mn>2</mn><mo>log</mo><mi>p</mi><mo>)</mo></mrow><mo>/</mo><mi>p</mi></mrow></msqrt><mo>,</mo><msqrt><mrow><mn>2</mn><mo>log</mo><mn>2</mn></mrow></msqrt><mo>)</mo></mrow></mrow></math></span> when <span><math><mi>p</mi></math></span> is sufficiently large as a function of <span><math><mi>β</mi></math></span>. Of special interest is the lower boundary of this interval which matches the large <span><math><mi>p</mi></math></span> asymptotics of the inverse temperature marking the hypothetical dynamical transition predicted in statistical physics. We show this as a consequence of a ‘soft’ version of the overlap gap property which asserts the existence of a distance gap of points of typical energy from a typical sample from the Gibbs measure. We further show that this latter property implies that stable algorithms seeking to return a point of at least typical energy are confined to an exponentially rare subset of that super-level set, provided that their success probability is not vanishingly small.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"192 ","pages":"Article 104792"},"PeriodicalIF":1.2,"publicationDate":"2025-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145528753","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 : 2025-10-24DOI: 10.1016/j.spa.2025.104813
Elena Bandini , Christian Keller
We study the optimal control of path-dependent piecewise deterministic processes. An appropriate dynamic programming principle is established. We prove that the associated value function is the unique minimax solution of the corresponding non-local path-dependent Hamilton–Jacobi–Bellman equation. This is the first well-posedness result for nonsmooth solutions of fully nonlinear non-local path-dependent partial differential equations.
{"title":"Non-local Hamilton–Jacobi–Bellman equations for the stochastic optimal control of path-dependent piecewise deterministic processes","authors":"Elena Bandini , Christian Keller","doi":"10.1016/j.spa.2025.104813","DOIUrl":"10.1016/j.spa.2025.104813","url":null,"abstract":"<div><div>We study the optimal control of path-dependent piecewise deterministic processes. An appropriate dynamic programming principle is established. We prove that the associated value function is the unique minimax solution of the corresponding non-local path-dependent Hamilton–Jacobi–Bellman equation. This is the first well-posedness result for nonsmooth solutions of fully nonlinear non-local path-dependent partial differential equations.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"192 ","pages":"Article 104813"},"PeriodicalIF":1.2,"publicationDate":"2025-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145418569","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 : 2025-10-24DOI: 10.1016/j.spa.2025.104810
Muneya Matsui , Thomas Mikosch , Olivier Wintenberger
We consider a regularly varyingstationary sequenceof random variables with tail index . For this sequence we study the joint convergenceof sums, -type moduli and maxima. We focus on ratio statistics, including the studentized sums and sums normalized by the corresponding maxima, and study the existence of moments for the limit ratios. We consider particular examples of processes whose limit ratios possess all moments as in the iid setting. But, in contrast to the latter situation, there also exist dependent sequences where certain moments of the limit ratio are infinite. This phenomenon results from extremal clusters in the sequence.
{"title":"Moments for self-normalized partial sums","authors":"Muneya Matsui , Thomas Mikosch , Olivier Wintenberger","doi":"10.1016/j.spa.2025.104810","DOIUrl":"10.1016/j.spa.2025.104810","url":null,"abstract":"<div><div>We consider a regularly varyingstationary sequenceof random variables <span><math><mrow><mo>(</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>)</mo></mrow></math></span> with tail index <span><math><mrow><mi>α</mi><mo><</mo><mn>2</mn></mrow></math></span>. For this sequence we study the joint convergenceof sums, <span><math><msup><mrow><mi>ℓ</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span>-type moduli and maxima. We focus on ratio statistics, including the studentized sums and sums normalized by the corresponding maxima, and study the existence of moments for the limit ratios. We consider particular examples of processes <span><math><mrow><mo>(</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>)</mo></mrow></math></span> whose limit ratios possess all moments as in the iid setting. But, in contrast to the latter situation, there also exist dependent sequences <span><math><mrow><mo>(</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>)</mo></mrow></math></span> where certain moments of the limit ratio are infinite. This phenomenon results from extremal clusters in the sequence.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"192 ","pages":"Article 104810"},"PeriodicalIF":1.2,"publicationDate":"2025-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145418568","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 : 2025-10-24DOI: 10.1016/j.spa.2025.104789
Rami Atar
This paper studies a branching-selection model of motionless particles in , with nonlocal branching, introduced by Durrett and Remenik in dimension 1. The assumptions on the fitness function, , and on the inhomogeneous branching distribution, are mild. The evolution equation for the macroscopic density is given by an integro-differential free boundary problem in , in which the free boundary represents the least -value in the population. The main result is the characterization of the limit in probability of the empirical measure process in terms of the unique solution to this free boundary problem.
{"title":"A Durrett–Remenik particle system in Rd","authors":"Rami Atar","doi":"10.1016/j.spa.2025.104789","DOIUrl":"10.1016/j.spa.2025.104789","url":null,"abstract":"<div><div>This paper studies a branching-selection model of motionless particles in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>, with nonlocal branching, introduced by Durrett and Remenik in dimension 1. The assumptions on the fitness function, <span><math><mi>F</mi></math></span>, and on the inhomogeneous branching distribution, are mild. The evolution equation for the macroscopic density is given by an integro-differential free boundary problem in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>, in which the free boundary represents the least <span><math><mi>F</mi></math></span>-value in the population. The main result is the characterization of the limit in probability of the empirical measure process in terms of the unique solution to this free boundary problem.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"192 ","pages":"Article 104789"},"PeriodicalIF":1.2,"publicationDate":"2025-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145528752","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 : 2025-10-23DOI: 10.1016/j.spa.2025.104812
G. Alsmeyer , F. Cordero , H. Dopmeyer
We consider a population with two types of individuals, distinguished by the resources required for reproduction: type-0 (small) individuals need a fractional resource unit of size , while type-1 (large) individuals require 1 unit. The total available resource per generation is . To form a new generation, individuals are sampled one by one, and if enough resources remain, they reproduce, adding their offspring to the next generation. The probability of sampling an individual whose offspring is small is , where is the proportion of small individuals in the current generation. We call this discrete-time stochastic model a two-size Wright–Fisher model, where the function can represent mutation and/or frequency-dependent selection. We show that on the evolutionary time scale, i.e. accelerating time by a factor , the frequency process of type-0 individuals converges to the solution of a Wright–Fisher-type SDE. The drift term of that SDE accounts for the bias introduced by the function and the consumption strategy, the latter also inducing an additional multiplicative factor in the diffusion term. To prove this, the dynamics within each generation are viewed as a renewal process, with the population size corresponding to the first passage time above level . The proof relies on methods from renewal theory, in particular a uniform version of Blackwell’s renewal theorem for binary, non-arithmetic random variables, established via -coupling.
{"title":"A two-size Wright–Fisher model: asymptotic analysis via uniform renewal theory","authors":"G. Alsmeyer , F. Cordero , H. Dopmeyer","doi":"10.1016/j.spa.2025.104812","DOIUrl":"10.1016/j.spa.2025.104812","url":null,"abstract":"<div><div>We consider a population with two types of individuals, distinguished by the resources required for reproduction: type-0 (small) individuals need a fractional resource unit of size <span><math><mrow><mi>ϑ</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span>, while type-1 (large) individuals require 1 unit. The total available resource per generation is <span><math><mi>R</mi></math></span>. To form a new generation, individuals are sampled one by one, and if enough resources remain, they reproduce, adding their offspring to the next generation. The probability of sampling an individual whose offspring is small is <span><math><mrow><msub><mrow><mi>ρ</mi></mrow><mrow><mi>R</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span>, where <span><math><mi>x</mi></math></span> is the proportion of small individuals in the current generation. We call this discrete-time stochastic model a two-size Wright–Fisher model, where the function <span><math><msub><mrow><mi>ρ</mi></mrow><mrow><mi>R</mi></mrow></msub></math></span> can represent mutation and/or frequency-dependent selection. We show that on the evolutionary time scale, i.e. accelerating time by a factor <span><math><mi>R</mi></math></span>, the frequency process of type-0 individuals converges to the solution of a Wright–Fisher-type SDE. The drift term of that SDE accounts for the bias introduced by the function <span><math><msub><mrow><mi>ρ</mi></mrow><mrow><mi>R</mi></mrow></msub></math></span> and the consumption strategy, the latter also inducing an additional multiplicative factor in the diffusion term. To prove this, the dynamics within each generation are viewed as a renewal process, with the population size corresponding to the first passage time <span><math><mrow><mi>τ</mi><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span> above level <span><math><mi>R</mi></math></span>. The proof relies on methods from renewal theory, in particular a uniform version of Blackwell’s renewal theorem for binary, non-arithmetic random variables, established via <span><math><mi>ɛ</mi></math></span>-coupling.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"192 ","pages":"Article 104812"},"PeriodicalIF":1.2,"publicationDate":"2025-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145418567","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}
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