Pub Date : 2025-02-27DOI: 10.1016/j.spa.2025.104614
Yacouba Boubacar Maïnassara , Landy Rabehasaina
We consider an observed subcritical Galton Watson process with correlated stationary immigration process . Two situations are presented. The first one is when for larger than some : a consistent estimator for the reproduction and mean immigration rates is given, and a central limit theorem is proved. The second one is when has general correlation structure: under mixing assumptions, we exhibit an estimator for the logarithm of the reproduction rate and we prove that it converges in quadratic mean with explicit speed. In addition, when the mixing coefficients decrease fast enough, we provide and prove a two terms expansion for the estimator. Numerical illustrations are provided.
{"title":"Estimation of subcritical Galton Watson processes with correlated immigration","authors":"Yacouba Boubacar Maïnassara , Landy Rabehasaina","doi":"10.1016/j.spa.2025.104614","DOIUrl":"10.1016/j.spa.2025.104614","url":null,"abstract":"<div><div>We consider an observed subcritical Galton Watson process <span><math><mrow><mo>{</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mspace></mspace><mi>n</mi><mo>∈</mo><mi>Z</mi><mo>}</mo></mrow></math></span> with correlated stationary immigration process <span><math><mrow><mo>{</mo><msub><mrow><mi>ϵ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mspace></mspace><mi>n</mi><mo>∈</mo><mi>Z</mi><mo>}</mo></mrow></math></span>. Two situations are presented. The first one is when <span><math><mrow><mtext>Cov</mtext><mrow><mo>(</mo><msub><mrow><mi>ϵ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>ϵ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow></math></span> for <span><math><mi>k</mi></math></span> larger than some <span><math><msub><mrow><mi>k</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>: a consistent estimator for the reproduction and mean immigration rates is given, and a central limit theorem is proved. The second one is when <span><math><mrow><mo>{</mo><msub><mrow><mi>ϵ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mspace></mspace><mi>n</mi><mo>∈</mo><mi>Z</mi><mo>}</mo></mrow></math></span> has general correlation structure: under mixing assumptions, we exhibit an estimator for the logarithm of the reproduction rate and we prove that it converges in quadratic mean with explicit speed. In addition, when the mixing coefficients decrease fast enough, we provide and prove a two terms expansion for the estimator. Numerical illustrations are provided.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"184 ","pages":"Article 104614"},"PeriodicalIF":1.1,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143526695","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 : 2025-02-25DOI: 10.1016/j.spa.2025.104612
Shiduo Qu, Hongjun Gao
This paper investigates a class of stochastic partial differential equations (SPDEs) driven by standard Brownian motion and fractional Brownian motion with Hurst parameter . We establish the existence and uniqueness of solutions for these SPDEs in sense of almost surely. We further prove that the moments of the solutions are finite. Moreover, we explore the equivalence between the integral defined by fractional derivatives and that defined by sewing lemma.
{"title":"Existence and uniqueness of SPDEs driven by nonlinear multiplicative mixed noise","authors":"Shiduo Qu, Hongjun Gao","doi":"10.1016/j.spa.2025.104612","DOIUrl":"10.1016/j.spa.2025.104612","url":null,"abstract":"<div><div>This paper investigates a class of stochastic partial differential equations (SPDEs) driven by standard Brownian motion and fractional Brownian motion with Hurst parameter <span><math><mrow><mi>H</mi><mo>></mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></math></span>. We establish the existence and uniqueness of solutions for these SPDEs in sense of almost surely. We further prove that the moments of the solutions are finite. Moreover, we explore the equivalence between the integral defined by fractional derivatives and that defined by sewing lemma.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"184 ","pages":"Article 104612"},"PeriodicalIF":1.1,"publicationDate":"2025-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143535025","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 : 2025-02-21DOI: 10.1016/j.spa.2025.104604
Tobias Boege , Mathias Drton , Benjamin Hollering , Sarah Lumpp , Pratik Misra , Daniela Schkoda
Stationary distributions of multivariate diffusion processes have recently been proposed as probabilistic models of causal systems in statistics and machine learning. Motivated by these developments, we study stationary multivariate diffusion processes with a sparsely structured drift. Our main result gives a characterization of the conditional independence relations that hold in a stationary distribution. The result draws on a graphical representation of the drift structure and pertains to conditional independence relations that hold generally as a consequence of the drift’s sparsity pattern.
{"title":"Conditional independence in stationary distributions of diffusions","authors":"Tobias Boege , Mathias Drton , Benjamin Hollering , Sarah Lumpp , Pratik Misra , Daniela Schkoda","doi":"10.1016/j.spa.2025.104604","DOIUrl":"10.1016/j.spa.2025.104604","url":null,"abstract":"<div><div>Stationary distributions of multivariate diffusion processes have recently been proposed as probabilistic models of causal systems in statistics and machine learning. Motivated by these developments, we study stationary multivariate diffusion processes with a sparsely structured drift. Our main result gives a characterization of the conditional independence relations that hold in a stationary distribution. The result draws on a graphical representation of the drift structure and pertains to conditional independence relations that hold generally as a consequence of the drift’s sparsity pattern.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"184 ","pages":"Article 104604"},"PeriodicalIF":1.1,"publicationDate":"2025-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143518982","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-14DOI: 10.1016/j.spa.2025.104603
Quyuan Lin , Rongchang Liu , Weinan Wang
In this paper, we consider the 2D stochastic Nernst–Planck–Navier–Stokes equations incorporating transport noise affecting both momentum and ionic concentrations. By assuming the ionic species have the same diffusivity and opposite valences, we prove the global well-posedness of the system. Furthermore, we illustrate the enhanced dissipation phenomenon in the system with specific transportation noise by establishing that it enables an arbitrarily large exponential convergence rate of the solutions.
{"title":"Global well-posedness and enhanced dissipation for the 2D stochastic Nernst–Planck–Navier–Stokes equations with transport noise","authors":"Quyuan Lin , Rongchang Liu , Weinan Wang","doi":"10.1016/j.spa.2025.104603","DOIUrl":"10.1016/j.spa.2025.104603","url":null,"abstract":"<div><div>In this paper, we consider the 2D stochastic Nernst–Planck–Navier–Stokes equations incorporating transport noise affecting both momentum and ionic concentrations. By assuming the ionic species have the same diffusivity and opposite valences, we prove the global well-posedness of the system. Furthermore, we illustrate the enhanced dissipation phenomenon in the system with specific transportation noise by establishing that it enables an arbitrarily large exponential convergence rate of the solutions.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"184 ","pages":"Article 104603"},"PeriodicalIF":1.1,"publicationDate":"2025-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143419924","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 : 2025-02-12DOI: 10.1016/j.spa.2025.104594
Mazyar Ghani Varzaneh , Sebastian Riedel , Alexander Schmeding , Nikolas Tapia
We prove that the spaces of controlled (branched) rough paths of arbitrary order form a continuous field of Banach spaces. This structure has many similarities to an (infinite-dimensional) vector bundle and allows to define a topology on the total space, the collection of all controlled path spaces, which turns out to be Polish in the geometric case. The construction is intrinsic and based on a new approximation result for controlled rough paths. This framework turns well-known maps such as the rough integration map and the Itô–Lyons map into continuous (structure preserving) mappings. Moreover, it is compatible with previous constructions of interest in the stability theory for rough integration.
{"title":"The geometry of controlled rough paths","authors":"Mazyar Ghani Varzaneh , Sebastian Riedel , Alexander Schmeding , Nikolas Tapia","doi":"10.1016/j.spa.2025.104594","DOIUrl":"10.1016/j.spa.2025.104594","url":null,"abstract":"<div><div>We prove that the spaces of controlled (branched) rough paths of arbitrary order form a continuous field of Banach spaces. This structure has many similarities to an (infinite-dimensional) vector bundle and allows to define a topology on the total space, the collection of all controlled path spaces, which turns out to be Polish in the geometric case. The construction is intrinsic and based on a new approximation result for controlled rough paths. This framework turns well-known maps such as the rough integration map and the Itô–Lyons map into continuous (structure preserving) mappings. Moreover, it is compatible with previous constructions of interest in the stability theory for rough integration.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"184 ","pages":"Article 104594"},"PeriodicalIF":1.1,"publicationDate":"2025-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143487789","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 : 2025-02-12DOI: 10.1016/j.spa.2025.104601
Gerardo Barrera , Conrado da-Costa , Milton Jara
In this paper, we study an ordinary differential equation with a degenerate global attractor at the origin, to which we add a white noise with a small parameter that regulates its intensity. Under general conditions, for any fixed intensity, as time tends to infinity, the solution of this stochastic dynamics converges exponentially fast in total variation distance to a unique equilibrium distribution. We suitably accelerate the random dynamics and show that the preceding convergence is gradual, that is, the function that associates to each fixed the total variation distance between the accelerated random dynamics at time and its equilibrium distribution converges, as the noise intensity tends to zero, to a decreasing function with values in . Moreover, we prove that this limit function for each fixed corresponds to the total variation distance between the marginal, at time , of a stochastic differential equation that comes down from infinity and its corresponding equilibrium distribution. This completes the classification of all possible behaviors of the total variation distance between the time marginal of the aforementioned stochastic dynamics and its invariant measure for one dimensional well-behaved convex potentials. In addition, there is no cut-off phenomenon for this one-parameter family of random processes and asymptotics of the mixing times are derived.
{"title":"Gradual convergence for Langevin dynamics on a degenerate potential","authors":"Gerardo Barrera , Conrado da-Costa , Milton Jara","doi":"10.1016/j.spa.2025.104601","DOIUrl":"10.1016/j.spa.2025.104601","url":null,"abstract":"<div><div>In this paper, we study an ordinary differential equation with a degenerate global attractor at the origin, to which we add a white noise with a small parameter that regulates its intensity. Under general conditions, for any fixed intensity, as time tends to infinity, the solution of this stochastic dynamics converges exponentially fast in total variation distance to a unique equilibrium distribution. We suitably accelerate the random dynamics and show that the preceding convergence is gradual, that is, the function that associates to each fixed <span><math><mrow><mi>t</mi><mo>≥</mo><mn>0</mn></mrow></math></span> the total variation distance between the accelerated random dynamics at time <span><math><mi>t</mi></math></span> and its equilibrium distribution converges, as the noise intensity tends to zero, to a decreasing function with values in <span><math><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></math></span>. Moreover, we prove that this limit function for each fixed <span><math><mrow><mi>t</mi><mo>≥</mo><mn>0</mn></mrow></math></span> corresponds to the total variation distance between the marginal, at time <span><math><mi>t</mi></math></span>, of a stochastic differential equation that comes down from infinity and its corresponding equilibrium distribution. This completes the classification of all possible behaviors of the total variation distance between the time marginal of the aforementioned stochastic dynamics and its invariant measure for one dimensional well-behaved convex potentials. In addition, there is no cut-off phenomenon for this one-parameter family of random processes and asymptotics of the mixing times are derived.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"184 ","pages":"Article 104601"},"PeriodicalIF":1.1,"publicationDate":"2025-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143419923","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-11DOI: 10.1016/j.spa.2025.104602
Hao Ding , Shizan Fang , Xiang-Dong Li
We establish the existence and uniqueness of stochastic parallel translations and diffusions driven by a Q-Wiener process on the Wasserstein space over . Surprisingly enough, the equation defining stochastic parallel translations is a SDE on a Hilbert space, instead of a SPDE.
{"title":"Stochastic parallel translations and diffusions on the Wasserstein space over T","authors":"Hao Ding , Shizan Fang , Xiang-Dong Li","doi":"10.1016/j.spa.2025.104602","DOIUrl":"10.1016/j.spa.2025.104602","url":null,"abstract":"<div><div>We establish the existence and uniqueness of stochastic parallel translations and diffusions driven by a Q-Wiener process on the Wasserstein space over <span><math><mi>T</mi></math></span>. Surprisingly enough, the equation defining stochastic parallel translations is a SDE on a Hilbert space, instead of a SPDE.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"184 ","pages":"Article 104602"},"PeriodicalIF":1.1,"publicationDate":"2025-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143419922","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 : 2025-02-10DOI: 10.1016/j.spa.2025.104599
Jorge Ignacio González Cázares , Feng Lin , Aleksandar Mijatović
This paper provides an exact simulation algorithm for the sampling from the joint law of the first-passage time, the undershoot and the overshoot of a subordinator crossing a non-increasing boundary. The algorithm applies to a large non-parametric class of subordinators of interest in applications. We prove that the running time of this algorithm has finite moments of all positive orders and give an explicit bound on the expected running time in terms of the Lévy measure of the subordinator. This bound provides performance guarantees that make our algorithm suitable for Monte Carlo estimation.
{"title":"Fast exact simulation of the first-passage event of a subordinator","authors":"Jorge Ignacio González Cázares , Feng Lin , Aleksandar Mijatović","doi":"10.1016/j.spa.2025.104599","DOIUrl":"10.1016/j.spa.2025.104599","url":null,"abstract":"<div><div>This paper provides an exact simulation algorithm for the sampling from the joint law of the first-passage time, the undershoot and the overshoot of a subordinator crossing a non-increasing boundary. The algorithm applies to a large non-parametric class of subordinators of interest in applications. We prove that the running time of this algorithm has finite moments of all positive orders and give an explicit bound on the expected running time in terms of the Lévy measure of the subordinator. This bound provides performance guarantees that make our algorithm suitable for Monte Carlo estimation.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"183 ","pages":"Article 104599"},"PeriodicalIF":1.1,"publicationDate":"2025-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143386638","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-10DOI: 10.1016/j.spa.2025.104600
Lu-Jing Huang , Tao Wang
In this paper, we establish the sufficient and necessary conditions for the symmetry of the following stable Lévy-type operator on : where is a continuous and strictly positive function, and is a differentiable function. We then study the criteria for functional inequalities, such as logarithmic Sobolev inequalities, Nash inequalities and super-Poincaré inequalities under the assumption of symmetry. Our approach involves the Orlicz space theory and the estimates of the Green functions.
{"title":"Symmetry and functional inequalities for stable Lévy-type operators","authors":"Lu-Jing Huang , Tao Wang","doi":"10.1016/j.spa.2025.104600","DOIUrl":"10.1016/j.spa.2025.104600","url":null,"abstract":"<div><div>In this paper, we establish the sufficient and necessary conditions for the symmetry of the following stable Lévy-type operator <span><math><mi>L</mi></math></span> on <span><math><mi>R</mi></math></span>: <span><span><span><math><mrow><mi>L</mi><mo>=</mo><mi>a</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msup><mrow><mi>Δ</mi></mrow><mrow><mi>α</mi><mo>/</mo><mn>2</mn></mrow></msup><mo>+</mo><mi>b</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mfrac><mrow><mi>d</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo>,</mo></mrow></math></span></span></span>where <span><math><mi>a</mi></math></span> is a continuous and strictly positive function, and <span><math><mi>b</mi></math></span> is a differentiable function. We then study the criteria for functional inequalities, such as logarithmic Sobolev inequalities, Nash inequalities and super-Poincaré inequalities under the assumption of symmetry. Our approach involves the Orlicz space theory and the estimates of the Green functions.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"183 ","pages":"Article 104600"},"PeriodicalIF":1.1,"publicationDate":"2025-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143395378","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 : 2025-02-10DOI: 10.1016/j.spa.2025.104597
Dariusz Buraczewski , Alexander Iksanov , Valeriya Kotelnikova
We prove a law of the iterated logarithm (LIL) for an infinite sum of independent indicators parameterized by and monotone in as . It is shown that if the expectation and the variance of the sum are comparable, then the normalization in the LIL includes the iterated logarithm of . If the expectation grows faster than the variance, while the ratio remains bounded, then the normalization in the LIL includes the single logarithm of (so that the LIL becomes a law of the single logarithm). Applications of our result are given to the number of points of the infinite Ginibre point process in a disk and the number of occupied boxes and related quantities in Karlin’s occupancy scheme.
{"title":"Laws of the iterated and single logarithm for sums of independent indicators, with applications to the Ginibre point process and Karlin’s occupancy scheme","authors":"Dariusz Buraczewski , Alexander Iksanov , Valeriya Kotelnikova","doi":"10.1016/j.spa.2025.104597","DOIUrl":"10.1016/j.spa.2025.104597","url":null,"abstract":"<div><div>We prove a law of the iterated logarithm (LIL) for an infinite sum of independent indicators parameterized by <span><math><mi>t</mi></math></span> and monotone in <span><math><mi>t</mi></math></span> as <span><math><mrow><mi>t</mi><mo>→</mo><mi>∞</mi></mrow></math></span>. It is shown that if the expectation <span><math><mi>b</mi></math></span> and the variance <span><math><mi>a</mi></math></span> of the sum are comparable, then the normalization in the LIL includes the iterated logarithm of <span><math><mi>a</mi></math></span>. If the expectation grows faster than the variance, while the ratio <span><math><mrow><mo>log</mo><mi>b</mi><mo>/</mo><mo>log</mo><mi>a</mi></mrow></math></span> remains bounded, then the normalization in the LIL includes the single logarithm of <span><math><mi>a</mi></math></span> (so that the LIL becomes a law of the single logarithm). Applications of our result are given to the number of points of the infinite Ginibre point process in a disk and the number of occupied boxes and related quantities in Karlin’s occupancy scheme.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"183 ","pages":"Article 104597"},"PeriodicalIF":1.1,"publicationDate":"2025-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143395379","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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