Let $W_{t}$ be Brownian motion in the plane started at the origin and let $ theta$ be the first exit time of the unit disk $D_{1}$. Let [mu_{ theta } ( x,epsilon) =frac{1}{piepsilon^{ 2} }int_{0}^{ theta }1_{{ B( x,epsilon)}}( W_{t}),dt,] and set $mu^{ ast}_{ theta } (epsilon)=sup_{xin D_{1}}mu_{ theta } ( x,epsilon)$. We show that [sqrt{mu^{ast}_{theta} (epsilon)}-sqrt{2/pi} left(log epsilon^{-1}- frac{1}{2}loglog epsilon^{-1}right)] is tight.
{"title":"Tightness for thick points in two dimensions","authors":"J. Rosen","doi":"10.1214/23-ejp910","DOIUrl":"https://doi.org/10.1214/23-ejp910","url":null,"abstract":"Let $W_{t}$ be Brownian motion in the plane started at the origin and let $ theta$ be the first exit time of the unit disk $D_{1}$. Let [mu_{ theta } ( x,epsilon) =frac{1}{piepsilon^{ 2} }int_{0}^{ theta }1_{{ B( x,epsilon)}}( W_{t}),dt,] and set $mu^{ ast}_{ theta } (epsilon)=sup_{xin D_{1}}mu_{ theta } ( x,epsilon)$. We show that [sqrt{mu^{ast}_{theta} (epsilon)}-sqrt{2/pi} left(log epsilon^{-1}- frac{1}{2}loglog epsilon^{-1}right)] is tight.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48828452","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Several long-time limit theorems of one-dimensional Lévy processes weighted and normalized by functions of the local time are studied. The long-time limits are taken via certain families of random times, called clocks: exponential clock, hitting time clock, two-point hitting time clock and inverse local time clock. The limit measure can be characterized via a certain martingale expressed by an invariant function for the process killed upon hitting zero. The limit processes may differ according to the choice of the clocks when the original Lévy process is recurrent and of finite variance.
{"title":"Local time penalizations with various clocks for Lévy processes","authors":"Shosei Takeda, K. Yano","doi":"10.1214/23-ejp903","DOIUrl":"https://doi.org/10.1214/23-ejp903","url":null,"abstract":"Several long-time limit theorems of one-dimensional Lévy processes weighted and normalized by functions of the local time are studied. The long-time limits are taken via certain families of random times, called clocks: exponential clock, hitting time clock, two-point hitting time clock and inverse local time clock. The limit measure can be characterized via a certain martingale expressed by an invariant function for the process killed upon hitting zero. The limit processes may differ according to the choice of the clocks when the original Lévy process is recurrent and of finite variance.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47058314","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the limits of the additive and derivative martingales of one-dimensional branching Brownian motion in a periodic environment. Then we prove the existence of pulsating travelling wave solutions of the corresponding F-KPP equation in the supercritical and critical cases by representing the solutions probabilistically in terms of the limits of the additive and derivative martingales. We also prove that there is no pulsating travelling wave solution in the subcritical case. Our main tools are the spine decomposition and martingale change of measures.
{"title":"Branching Brownian motion in a periodic environment and existence of pulsating traveling waves","authors":"Y-X. Ren, R. Song, Fan Yang","doi":"10.1214/23-ejp960","DOIUrl":"https://doi.org/10.1214/23-ejp960","url":null,"abstract":"We study the limits of the additive and derivative martingales of one-dimensional branching Brownian motion in a periodic environment. Then we prove the existence of pulsating travelling wave solutions of the corresponding F-KPP equation in the supercritical and critical cases by representing the solutions probabilistically in terms of the limits of the additive and derivative martingales. We also prove that there is no pulsating travelling wave solution in the subcritical case. Our main tools are the spine decomposition and martingale change of measures.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45044438","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove time-dependent versions of Kingman's subadditive ergodic theorem, which can be used to study stochastic processes as well as propagation of solutions to PDE in time-dependent environments.
{"title":"Subadditive theorems in time-dependent environments","authors":"Yuming Zhang, Andrej Zlatoš","doi":"10.1214/23-ejp990","DOIUrl":"https://doi.org/10.1214/23-ejp990","url":null,"abstract":"We prove time-dependent versions of Kingman's subadditive ergodic theorem, which can be used to study stochastic processes as well as propagation of solutions to PDE in time-dependent environments.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46974226","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove the convergence of a class of high frequency path-functionals of a sticky diffusion to its local time. First, we prove this for the sticky Brownian motion. Then, we extend the result to sticky stochastic differential equations. We combine the local time approximation with an approximation of the occupation time to set up a consistent stickiness estimator. Last, we perform numerical experiments to assess the properties of the stickiness estimator and the local time approximation.
{"title":"Functional convergence to the local time of a sticky diffusion","authors":"Alexis Anagnostakis","doi":"10.1214/23-ejp972","DOIUrl":"https://doi.org/10.1214/23-ejp972","url":null,"abstract":"We prove the convergence of a class of high frequency path-functionals of a sticky diffusion to its local time. First, we prove this for the sticky Brownian motion. Then, we extend the result to sticky stochastic differential equations. We combine the local time approximation with an approximation of the occupation time to set up a consistent stickiness estimator. Last, we perform numerical experiments to assess the properties of the stickiness estimator and the local time approximation.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44569516","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider a Fisher-KPP equation with nonlinear selection driven by a Poisson random measure. We prove that the equation admits a unique wave speed $ mathfrak{s}>0 $ given by $frac{mathfrak{s}^{2}}{2} = int_{[0, 1]}frac{ log{(1 + y)}}{y} mathfrak{R}( mathrm d y)$ where $ mathfrak{R} $ is the intensity of the impacts of the driving noise. Our arguments are based on upper and lower bounds via a quenched duality with a coordinated system of branching Brownian motions.
我们考虑由泊松随机测度驱动的具有非线性选择的Fisher KPP方程。我们证明了该方程允许由$frac｛mathfrak｛s｝^｛2｝｝｛2}=int_{[0，1]｝frac{log｛（1+y）｝{y｝mathfrak｛R｝（mathrm d y）$给出的唯一波速$mathfrak{s｝>0$，其中$mathfrak{R｝$是驾驶噪声影响的强度。我们的论点是基于上界和下界，通过具有分支布朗运动协调系统的猝灭对偶。
{"title":"The wave speed of an FKPP equation with jumps via coordinated branching","authors":"T. Rosati, Andr'as T'obi'as","doi":"10.1214/23-ejp958","DOIUrl":"https://doi.org/10.1214/23-ejp958","url":null,"abstract":"We consider a Fisher-KPP equation with nonlinear selection driven by a Poisson random measure. We prove that the equation admits a unique wave speed $ mathfrak{s}>0 $ given by $frac{mathfrak{s}^{2}}{2} = int_{[0, 1]}frac{ log{(1 + y)}}{y} mathfrak{R}( mathrm d y)$ where $ mathfrak{R} $ is the intensity of the impacts of the driving noise. Our arguments are based on upper and lower bounds via a quenched duality with a coordinated system of branching Brownian motions.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47631801","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider a stochastic individual-based population model with competition, trait-structure affecting reproduction and survival, and changing environment. The changes of traits are described by jump processes, and the dynamics can be approximated in large population by a non-linear PDE with a non-local mutation operator. Using the fact that this PDE admits a non-trivial stationary solution, we can approximate the non-linear stochastic population process by a linear birth-death process where the interactions are frozen, as long as the population remains close to this equilibrium. This allows us to derive, when the population is large, the equation satisfied by the ancestral lineage of an individual uniformly sampled at a fixed time $T$, which is the path constituted of the traits of the ancestors of this individual in past times $tleq T$. This process is a time inhomogeneous Markov process, but we show that the time reversal of this process possesses a very simple structure (e.g. time-homogeneous and independent of $T$). This extends recent results where the authors studied a similar model with a Laplacian operator but where the methods essentially relied on the Gaussian nature of the mutations.
{"title":"Time reversal of spinal processes for linear and non-linear branching processes near stationarity","authors":"Benoît Henry, S. M'el'eard, V. Tran","doi":"10.1214/23-ejp911","DOIUrl":"https://doi.org/10.1214/23-ejp911","url":null,"abstract":"We consider a stochastic individual-based population model with competition, trait-structure affecting reproduction and survival, and changing environment. The changes of traits are described by jump processes, and the dynamics can be approximated in large population by a non-linear PDE with a non-local mutation operator. Using the fact that this PDE admits a non-trivial stationary solution, we can approximate the non-linear stochastic population process by a linear birth-death process where the interactions are frozen, as long as the population remains close to this equilibrium. This allows us to derive, when the population is large, the equation satisfied by the ancestral lineage of an individual uniformly sampled at a fixed time $T$, which is the path constituted of the traits of the ancestors of this individual in past times $tleq T$. This process is a time inhomogeneous Markov process, but we show that the time reversal of this process possesses a very simple structure (e.g. time-homogeneous and independent of $T$). This extends recent results where the authors studied a similar model with a Laplacian operator but where the methods essentially relied on the Gaussian nature of the mutations.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41947489","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the Generalized Gibbs ensemble of the Ablowitz-Ladik lattice, and the Schur flow. We derive large deviations principles for the distribution of the empirical measures of the equilibrium measures for these ensembles. As a consequence, we deduce their almost sure convergence. Moreover, we are able to characterize their limit in terms of the equilibrium measure of the Circular, and the Jacobi beta ensemble respectively.
{"title":"Large deviations for Ablowitz-Ladik lattice, and the Schur flow","authors":"Guido Mazzuca, Ronan Memin","doi":"10.1214/23-EJP941","DOIUrl":"https://doi.org/10.1214/23-EJP941","url":null,"abstract":"We consider the Generalized Gibbs ensemble of the Ablowitz-Ladik lattice, and the Schur flow. We derive large deviations principles for the distribution of the empirical measures of the equilibrium measures for these ensembles. As a consequence, we deduce their almost sure convergence. Moreover, we are able to characterize their limit in terms of the equilibrium measure of the Circular, and the Jacobi beta ensemble respectively.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42646790","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we study the joint behaviour of the degree, depth and label of and graph distance between high-degree vertices in the random recursive tree. We generalise the results obtained by Eslava and extend these to include the labels of and graph distance between high-degree vertices. The analysis of both these two properties of high-degree vertices is novel, in particular in relation to the behaviour of the depth of such vertices. In passing, we also obtain results for the joint behaviour of the degree and depth of and graph distance between any fixed number of vertices with a prescribed label. This combines several isolated results on the degree and depth of and graph distance between vertices with a prescribed label already present in the literature. Furthermore, we extend these results to hold jointly for any number of fixed vertices and improve these results by providing more detailed descriptions of the distributional limits. Our analysis is based on a correspondence between the random recursive tree and a representation of the Kingman $n$-coalescent.
{"title":"On joint properties of vertices with a given degree or label in the random recursive tree","authors":"B. Lodewijks","doi":"10.1214/22-ejp877","DOIUrl":"https://doi.org/10.1214/22-ejp877","url":null,"abstract":"In this paper, we study the joint behaviour of the degree, depth and label of and graph distance between high-degree vertices in the random recursive tree. We generalise the results obtained by Eslava and extend these to include the labels of and graph distance between high-degree vertices. The analysis of both these two properties of high-degree vertices is novel, in particular in relation to the behaviour of the depth of such vertices. In passing, we also obtain results for the joint behaviour of the degree and depth of and graph distance between any fixed number of vertices with a prescribed label. This combines several isolated results on the degree and depth of and graph distance between vertices with a prescribed label already present in the literature. Furthermore, we extend these results to hold jointly for any number of fixed vertices and improve these results by providing more detailed descriptions of the distributional limits. Our analysis is based on a correspondence between the random recursive tree and a representation of the Kingman $n$-coalescent.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43560161","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we show that the diagonal of a high-dimensional sample covariance matrix stemming from $n$ independent observations of a $p$-dimensional time series with finite fourth moments can be approximated in spectral norm by the diagonal of the population covariance matrix. We assume that $n,pto infty$ with $p/n$ tending to a constant which might be positive or zero. As applications, we provide an approximation of the sample correlation matrix ${mathbf R}$ and derive a variety of results for its eigenvalues. We identify the limiting spectral distribution of ${mathbf R}$ and construct an estimator for the population correlation matrix and its eigenvalues. Finally, the almost sure limits of the extreme eigenvalues of ${mathbf R}$ in a generalized spiked correlation model are analyzed.
{"title":"Large sample correlation matrices: a comparison theorem and its applications","authors":"J. Heiny","doi":"10.1214/22-ejp817","DOIUrl":"https://doi.org/10.1214/22-ejp817","url":null,"abstract":"In this paper, we show that the diagonal of a high-dimensional sample covariance matrix stemming from $n$ independent observations of a $p$-dimensional time series with finite fourth moments can be approximated in spectral norm by the diagonal of the population covariance matrix. We assume that $n,pto infty$ with $p/n$ tending to a constant which might be positive or zero. As applications, we provide an approximation of the sample correlation matrix ${mathbf R}$ and derive a variety of results for its eigenvalues. We identify the limiting spectral distribution of ${mathbf R}$ and construct an estimator for the population correlation matrix and its eigenvalues. Finally, the almost sure limits of the extreme eigenvalues of ${mathbf R}$ in a generalized spiked correlation model are analyzed.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44124562","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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