We consider the branching process in random environment ${Z_n}_{ngeq 0}$, which is a~population growth process where individuals reproduce independently of each other with the reproduction law randomly picked at each generation. We focus on the supercritical case, when the process survives with positive probability and grows exponentially fast on the nonextinction set. Using Fourier techniques we obtain Edgeworth expansions and the renewal theorem for the sequence ${log Z_n}_{nge 0}$ as well as we essentially improve the central limit theorem. Our strategy is to compare $log Z_n$ with partial sums of i.i.d. random variables in order to obtain precise estimates.
{"title":"Limit theorems for supercritical branching processes in random environment","authors":"D. Buraczewski, E. Damek","doi":"10.3150/21-bej1349","DOIUrl":"https://doi.org/10.3150/21-bej1349","url":null,"abstract":"We consider the branching process in random environment ${Z_n}_{ngeq 0}$, which is a~population growth process where individuals reproduce independently of each other with the reproduction law randomly picked at each generation. We focus on the supercritical case, when the process survives with positive probability and grows exponentially fast on the nonextinction set. Using Fourier techniques we obtain Edgeworth expansions and the renewal theorem for the sequence ${log Z_n}_{nge 0}$ as well as we essentially improve the central limit theorem. Our strategy is to compare $log Z_n$ with partial sums of i.i.d. random variables in order to obtain precise estimates.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"62 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84250159","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
By using the heat kernel parameter expansion with respect to the frozen SDEs, the intrinsic derivative is estimated for the law of Mckean-Vlasov SDEs with respect to the initial distribution. As an application, the total variation distance between the laws of two solutions is bounded by the Wasserstein distance for initial distributions. These extend some recent results proved for distribution-free noise by using the coupling method and Malliavin calculus.
{"title":"Derivative estimates on distributions of McKean-Vlasov SDEs","authors":"Xing Huang, Feng-Yu Wang","doi":"10.1214/21-EJP582","DOIUrl":"https://doi.org/10.1214/21-EJP582","url":null,"abstract":"By using the heat kernel parameter expansion with respect to the frozen SDEs, the intrinsic derivative is estimated for the law of Mckean-Vlasov SDEs with respect to the initial distribution. As an application, the total variation distance between the laws of two solutions is bounded by the Wasserstein distance for initial distributions. These extend some recent results proved for distribution-free noise by using the coupling method and Malliavin calculus.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"22 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88951995","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For a family of random-cluster models with cluster weights $qgeq 1$, we prove that the probability that $0$ is connected to $x$ is asymptotically equal to $tfrac{1}{q}chi(beta)^{2}beta J_{0,x}$. The method developed in this article can be applied to any spin model for which there exists a random-cluster representation which is one-monotonic.
{"title":"Sharp asymptotics of correlation functions in the subcritical long-range random-cluster and Potts models","authors":"Y. Aoun","doi":"10.1214/21-ECP390","DOIUrl":"https://doi.org/10.1214/21-ECP390","url":null,"abstract":"For a family of random-cluster models with cluster weights $qgeq 1$, we prove that the probability that $0$ is connected to $x$ is asymptotically equal to $tfrac{1}{q}chi(beta)^{2}beta J_{0,x}$. The method developed in this article can be applied to any spin model for which there exists a random-cluster representation which is one-monotonic.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"57 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82305699","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that quantum Schur-Weyl duality leads to Markov duality for a variety of asymmetric interacting particle systems. In particular, we consider three cases: (1) Using a Schur-Weyl duality between a two-parameter quantum group and a two-parameter Hecke algebra from arXiv:math/0108038, we recover the Markov self-duality of multi-species ASEP previously discovered in arXiv:1605.00691 and arXiv:1606.04587. (2) From a Schur-Weyl duality between a co-ideal subalgebra of a quantum group and a Hecke algebra of type B arXiv:1609.01766, we find a Markov duality for a multi-species open ASEP on the semi-infinite line. The duality functional has not previously appeared in the literature. (3) A "fused" Hecke algebra from arXiv:2001.11372 leads to a new process, which we call braided ASEP. In braided ASEP, up to m particles may occupy a site and up to m particles may jump at a time. The Schur-Weyl duality between this Hecke algebra and a quantum group lead to a Markov duality. The duality function had previously appeared as the duality function of the multi-species ASEP(q,m/2) arXiv:1605.00691 and the stochastic multi-species higher spin vertex model arXiv:1701.04468.
{"title":"Two Dualities: Markov and Schur–Weyl","authors":"Jeffrey Kuan","doi":"10.1093/IMRN/RNAA333","DOIUrl":"https://doi.org/10.1093/IMRN/RNAA333","url":null,"abstract":"We show that quantum Schur-Weyl duality leads to Markov duality for a variety of asymmetric interacting particle systems. In particular, we consider three cases: \u0000(1) Using a Schur-Weyl duality between a two-parameter quantum group and a two-parameter Hecke algebra from arXiv:math/0108038, we recover the Markov self-duality of multi-species ASEP previously discovered in arXiv:1605.00691 and arXiv:1606.04587. \u0000(2) From a Schur-Weyl duality between a co-ideal subalgebra of a quantum group and a Hecke algebra of type B arXiv:1609.01766, we find a Markov duality for a multi-species open ASEP on the semi-infinite line. The duality functional has not previously appeared in the literature. \u0000(3) A \"fused\" Hecke algebra from arXiv:2001.11372 leads to a new process, which we call braided ASEP. In braided ASEP, up to m particles may occupy a site and up to m particles may jump at a time. The Schur-Weyl duality between this Hecke algebra and a quantum group lead to a Markov duality. The duality function had previously appeared as the duality function of the multi-species ASEP(q,m/2) arXiv:1605.00691 and the stochastic multi-species higher spin vertex model arXiv:1701.04468.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"56 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74438725","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we investigate the local limit theorem for additive functionals of a nonstationary Markov chain with finite or infinite second moment. The moment conditions are imposed on the individual summands and the weak dependence structure is expressed in terms of some uniformly mixing coefficients.
{"title":"On the local limit theorems for psi-mixing Markov chains","authors":"F. Merlevède, M. Peligrad, C. Peligrad","doi":"10.30757/alea.v18-45","DOIUrl":"https://doi.org/10.30757/alea.v18-45","url":null,"abstract":"In this paper we investigate the local limit theorem for additive functionals of a nonstationary Markov chain with finite or infinite second moment. The moment conditions are imposed on the individual summands and the weak dependence structure is expressed in terms of some uniformly mixing coefficients.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"05 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88360628","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We construct irreducible balanced non-transitive sets of n-sided dice for any positive integer n. We also study so-called a fair set of dice, which is one main tool in the construction of irreducible balanced non-transitive sets of dice. Furthermore, we study possible probabilities of non-transitive sets of dice.
{"title":"Possible Probability and Irreducibility of Balanced Nontransitive Dice","authors":"Injo Hur, Yeansu Kim","doi":"10.1155/2021/6648248","DOIUrl":"https://doi.org/10.1155/2021/6648248","url":null,"abstract":"We construct irreducible balanced non-transitive sets of n-sided dice for any positive integer n. We also study so-called a fair set of dice, which is one main tool in the construction of irreducible balanced non-transitive sets of dice. Furthermore, we study possible probabilities of non-transitive sets of dice.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"74 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77408570","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We find the best approximation of the fractional Brownian motion with the Hurst index $Hin (0,1/2)$ by Gaussian martingales of the form $int _0^ts^{gamma}dW_s$, where $W$ is a Wiener process, $gamma >0$.
{"title":"Distance from fractional Brownian motion with associated Hurst index 0<</mo>H<</mo>1/2 to the subspaces of Gaussian martingales involving power integrands with an arbitrary positive exponent","authors":"O. Banna, Filipp Buryak, Y. Mishura","doi":"10.15559/20-VMSTA156","DOIUrl":"https://doi.org/10.15559/20-VMSTA156","url":null,"abstract":"We find the best approximation of the fractional Brownian motion with the Hurst index $Hin (0,1/2)$ by Gaussian martingales of the form $int _0^ts^{gamma}dW_s$, where $W$ is a Wiener process, $gamma >0$.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"5 1","pages":"191-202"},"PeriodicalIF":0.0,"publicationDate":"2020-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89802026","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the action of Mandelbrot multiplicative cascades on the probability measures supported on a symbolic space, especially the ergodic measures. For general probability measures, we obtain almost a sharp criterion of non-degeneracy of the limiting measure; it relies on the lower and upper Hausdorff dimensions of the measure and the entropy of the weights generating the cascade. We also obtain sharp general bounds for the lower Hausdorff and upper packing dimensions of the limiting random measure when it is non-degenerate. When the original measure is a Gibbs measure associated with a measurable potential, all our results are sharp. This improves on results previously obtained by Kahane and Peyriere, Ben Nasr, and Fan, who considered the case of Markov measures. We exploit our results on general measures to derive dimensions estimates for some random measures on Bedford-McMullen carpets, as well as absolute continuity properties, with respect to their expectation, of the projections of some random statistically self-similar measures.
{"title":"On the Action of Multiplicative Cascades on Measures","authors":"J. Barral, Xiong Jin","doi":"10.1093/IMRN/RNAB125","DOIUrl":"https://doi.org/10.1093/IMRN/RNAB125","url":null,"abstract":"We consider the action of Mandelbrot multiplicative cascades on the probability measures supported on a symbolic space, especially the ergodic measures. For general probability measures, we obtain almost a sharp criterion of non-degeneracy of the limiting measure; it relies on the lower and upper Hausdorff dimensions of the measure and the entropy of the weights generating the cascade. We also obtain sharp general bounds for the lower Hausdorff and upper packing dimensions of the limiting random measure when it is non-degenerate. When the original measure is a Gibbs measure associated with a measurable potential, all our results are sharp. This improves on results previously obtained by Kahane and Peyriere, Ben Nasr, and Fan, who considered the case of Markov measures. We exploit our results on general measures to derive dimensions estimates for some random measures on Bedford-McMullen carpets, as well as absolute continuity properties, with respect to their expectation, of the projections of some random statistically self-similar measures.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"58 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91137953","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Motivated by the goal of understanding the evolution of populations undergoing selection, we consider branching Brownian motion in which particles independently move according to one-dimensional Brownian motion with drift, each particle may either split into two or die, and the difference between the birth and death rates is a linear function of the position of the particle. We show that, under certain assumptions, after a sufficiently long time, the empirical distribution of the positions of the particles is approximately Gaussian. This provides mathematically rigorous justification for results in the biology literature indicating that the distribution of the fitness levels of individuals in a population over time evolves like a Gaussian traveling wave.
{"title":"A Gaussian particle distribution for branching Brownian motion with an inhomogeneous branching rate","authors":"Matthew I. Roberts, Jason Schweinsberg","doi":"10.1214/21-ejp673","DOIUrl":"https://doi.org/10.1214/21-ejp673","url":null,"abstract":"Motivated by the goal of understanding the evolution of populations undergoing selection, we consider branching Brownian motion in which particles independently move according to one-dimensional Brownian motion with drift, each particle may either split into two or die, and the difference between the birth and death rates is a linear function of the position of the particle. We show that, under certain assumptions, after a sufficiently long time, the empirical distribution of the positions of the particles is approximately Gaussian. This provides mathematically rigorous justification for results in the biology literature indicating that the distribution of the fitness levels of individuals in a population over time evolves like a Gaussian traveling wave.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"28 3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72697370","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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