Pub Date : 2016-07-05DOI: 10.19195/0208-4147.38.2.2
Atef Lechiheb, I. Nourdin, Guangqu Zheng, Ezedine Haouala
This paper deals with the asymptotic behavior of random oscillatory integrals in the presence of long-range dependence. As a byproduct, we solve the corrector problem in random homogenization of onedimensional elliptic equations with highly oscillatory random coefficients displaying long-range dependence, by proving convergence to stochastic integrals with respect to Hermite processes.
{"title":"Convergence of random oscillatory integrals in the presence of long-range dependence and application to homogenization","authors":"Atef Lechiheb, I. Nourdin, Guangqu Zheng, Ezedine Haouala","doi":"10.19195/0208-4147.38.2.2","DOIUrl":"https://doi.org/10.19195/0208-4147.38.2.2","url":null,"abstract":"This paper deals with the asymptotic behavior of random oscillatory integrals in the presence of long-range dependence. As a byproduct, we solve the corrector problem in random homogenization of onedimensional elliptic equations with highly oscillatory random coefficients displaying long-range dependence, by proving convergence to stochastic integrals with respect to Hermite processes.","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2016-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68001053","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2016-06-24DOI: 10.19195/0208-4147.39.1.10
A. Genadot
In this paper, a class of piecewise deterministic Markov processes with underlying fast dynamic is studied. By using a “penalty method”, an averaging result is obtained when the underlying dynamic is infinitely accelerated. The features of the averaged process, which is still a piecewise deterministic Markov process, are fully described.
{"title":"Averaging for some simple constrained Markov processes","authors":"A. Genadot","doi":"10.19195/0208-4147.39.1.10","DOIUrl":"https://doi.org/10.19195/0208-4147.39.1.10","url":null,"abstract":"In this paper, a class of piecewise deterministic Markov processes with underlying fast dynamic is studied. By using a “penalty method”, an averaging result is obtained when the underlying dynamic is infinitely accelerated. The features of the averaged process, which is still a piecewise deterministic Markov process, are fully described.","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2016-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68001693","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2016-06-09DOI: 10.19195/0208-4147.38.1.9
K. Hees, H. Scheffler
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{"title":"On joint sum/max stability and sum/max domains of attraction","authors":"K. Hees, H. Scheffler","doi":"10.19195/0208-4147.38.1.9","DOIUrl":"https://doi.org/10.19195/0208-4147.38.1.9","url":null,"abstract":"Tu wpisz tekst","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2016-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68000898","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2016-05-06DOI: 10.19195/0208-4147.38.1.7
Selim Bahadır, E. Ceyhan
For a random sample of points in R, we consider the number of pairs whose members are nearest neighbors NNs to each other and the number of pairs sharing a common NN. The pairs of the first type are called reflexive NNs, whereas the pairs of the latter type are called shared NNs. In this article, we consider the case where the random sample of size n is from the uniform distribution on an interval. We denote the number of reflexive NN pairs and the number of shared NN pairs in the sample by Rn and Qn, respectively. We derive the exact forms of the expected value and the variance for both Rn and Qn, and derive a recurrence relation for Rn which may also be used to compute the exact probability mass function pmf of Rn. Our approach is a novel method for finding the pmf of Rn and agrees with the results in the literature. We also present SLLN and CLT results for both Rn and Qn as n goes to infinity.
{"title":"On the number of reflexive and shared nearest neighbor pairs in one-dimensional uniform data","authors":"Selim Bahadır, E. Ceyhan","doi":"10.19195/0208-4147.38.1.7","DOIUrl":"https://doi.org/10.19195/0208-4147.38.1.7","url":null,"abstract":"For a random sample of points in R, we consider the number of pairs whose members are nearest neighbors NNs to each other and the number of pairs sharing a common NN. The pairs of the first type are called reflexive NNs, whereas the pairs of the latter type are called shared NNs. In this article, we consider the case where the random sample of size n is from the uniform distribution on an interval. We denote the number of reflexive NN pairs and the number of shared NN pairs in the sample by Rn and Qn, respectively. We derive the exact forms of the expected value and the variance for both Rn and Qn, and derive a recurrence relation for Rn which may also be used to compute the exact probability mass function pmf of Rn. Our approach is a novel method for finding the pmf of Rn and agrees with the results in the literature. We also present SLLN and CLT results for both Rn and Qn as n goes to infinity.","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2016-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68000572","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-06-14DOI: 10.19195/0208-4147.38.2.6
H. Kosters, A. Tikhomirov
For fixed l≥0 and m≥1, let Xn0, Xn1,..., Xnl be independent random n × n matrices with independent entries, let Fn0 := Xn0, Xn1-1,..., Xnl-1, and let Fn1,..., Fnm be independent random matrices of the same form as Fn0 . We show that as n → ∞, the matrices Fn0 and m−l+1/2Fn1 +...+ Fnm have the same limiting eigenvalue distribution. To obtain our results, we apply the general framework recently introduced in Götze, Kösters, and Tikhomirov 2015 to sums of products of independent random matrices and their inverses.We establish the universality of the limiting singular value and eigenvalue distributions, and we provide a closer description of the limiting distributions in terms of free probability theory.
{"title":"Limiting spectral distributions of sums of products of non-Hermitian random matrices","authors":"H. Kosters, A. Tikhomirov","doi":"10.19195/0208-4147.38.2.6","DOIUrl":"https://doi.org/10.19195/0208-4147.38.2.6","url":null,"abstract":"For fixed l≥0 and m≥1, let Xn0, Xn1,..., Xnl be independent random n × n matrices with independent entries, let Fn0 := Xn0, Xn1-1,..., Xnl-1, and let Fn1,..., Fnm be independent random matrices of the same form as Fn0 . We show that as n → ∞, the matrices Fn0 and m−l+1/2Fn1 +...+ Fnm have the same limiting eigenvalue distribution. To obtain our results, we apply the general framework recently introduced in Götze, Kösters, and Tikhomirov 2015 to sums of products of independent random matrices and their inverses.We establish the universality of the limiting singular value and eigenvalue distributions, and we provide a closer description of the limiting distributions in terms of free probability theory.","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2015-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68001572","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-04-12DOI: 10.19195/0208-4147.38.2.4
Christoph Kopp, I. Molchanov
A stochastically continuous process ξt, t≥0, is said to be time-stable if the sum of n i.i.d. copies of ξ equals in distribution the time-scaled stochastic process ξnt, t≥0. The paper advances the understanding of time-stable processes by means of their LePage series representations as the sum of i.i.d. processes with the arguments scaled by the sequence of successive points of the unit intensity Poisson process on [0;∞. These series yield numerous examples of stochastic processes that share one-dimensional distributions with a Lévy process.
随机连续过程ξt, t≥-0，如果ξ的n i id个副本的和在时间尺度随机过程ξnt, t≥0的分布上相等，则称为时间稳定过程。本文通过将时间稳定过程的LePage级数表示为i.i.d过程的和，其参数由单位强度泊松过程在[0;∞上的连续点的序列缩放，提出了时间稳定过程的理解。这些序列产生了许多与lsamvy过程共享一维分布的随机过程的例子。
{"title":"Series representation of time-stable stochastic processes","authors":"Christoph Kopp, I. Molchanov","doi":"10.19195/0208-4147.38.2.4","DOIUrl":"https://doi.org/10.19195/0208-4147.38.2.4","url":null,"abstract":"A stochastically continuous process ξt, t≥0, is said to be time-stable if the sum of n i.i.d. copies of ξ equals in distribution the time-scaled stochastic process ξnt, t≥0. The paper advances the understanding of time-stable processes by means of their LePage series representations as the sum of i.i.d. processes with the arguments scaled by the sequence of successive points of the unit intensity Poisson process on [0;∞. These series yield numerous examples of stochastic processes that share one-dimensional distributions with a Lévy process.","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2015-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68000660","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2014-09-30DOI: 10.19195/0208-4147.38.1.2
Julien Letemplier, T. Simon
A multiplicative identity in law for the area of a spectrally positive Lévy ∝-stable process stopped at zero is established. Extending that of Lefebvre for Brownian motion, it involves an inverse beta random variable and the square of a positive stable random variable. This simple identity makes it possible to study precisely the behaviour of the density at zero, which is Fréchet-like.
{"title":"The area of a spectrally positive stable process stopped at zero","authors":"Julien Letemplier, T. Simon","doi":"10.19195/0208-4147.38.1.2","DOIUrl":"https://doi.org/10.19195/0208-4147.38.1.2","url":null,"abstract":"A multiplicative identity in law for the area of a spectrally positive Lévy ∝-stable process stopped at zero is established. Extending that of Lefebvre for Brownian motion, it involves an inverse beta random variable and the square of a positive stable random variable. This simple identity makes it possible to study precisely the behaviour of the density at zero, which is Fréchet-like.","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2014-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68000855","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2014-09-16DOI: 10.19195/0208-4147.39.2.2
Björn Böttcher
We prove a J1-tightness condition for embedded Markov chains and discuss four Skorokhod topologies in a unified manner. To approximate a continuous time stochastic process by discrete time Markov chains, one has several options to embed the Markov chains into continuous time processes. On the one hand, there is a Markov embedding which uses exponential waiting times. On the other hand, each Skorokhod topology naturally suggests a certain embedding. These are the step function embedding for J1, the linear interpolation embedding forM1, the multistep embedding for J2 and a more general embedding for M2. We show that the convergence of the step function embedding in J1 implies the convergence of the other embeddings in the corresponding topologies. For the converse statement, a J1-tightness condition for embedded time-homogeneous Markov chains is given.Additionally, it is shown that J1 convergence is equivalent to the joint convergence in M1 and J2.
{"title":"Embedded Markov chain approximations in Skorokhod topologies","authors":"Björn Böttcher","doi":"10.19195/0208-4147.39.2.2","DOIUrl":"https://doi.org/10.19195/0208-4147.39.2.2","url":null,"abstract":"We prove a J1-tightness condition for embedded Markov chains and discuss four Skorokhod topologies in a unified manner. To approximate a continuous time stochastic process by discrete time Markov chains, one has several options to embed the Markov chains into continuous time processes. On the one hand, there is a Markov embedding which uses exponential waiting times. On the other hand, each Skorokhod topology naturally suggests a certain embedding. These are the step function embedding for J1, the linear interpolation embedding forM1, the multistep embedding for J2 and a more general embedding for M2. We show that the convergence of the step function embedding in J1 implies the convergence of the other embeddings in the corresponding topologies. For the converse statement, a J1-tightness condition for embedded time-homogeneous Markov chains is given.Additionally, it is shown that J1 convergence is equivalent to the joint convergence in M1 and J2.","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2014-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68001893","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2013-06-11DOI: 10.19195/0208-4147.38.1.5
P. Vellaisamy, A. Maheshwari
In this paper, we define a fractional negative binomial process FNBP by replacing the Poisson process by a fractional Poisson process FPP in the gamma subordinated form of the negative binomial process. It is shown that the one-dimensional distributions of the FPP and the FNBP are not infinitely divisible. Also, the space fractional Pólya process SFPP is defined by replacing the rate parameter λ by a gamma random variable in the definition of the space fractional Poisson process. The properties of the FNBP and the SFPP and the connections to PDEs governing the density of the FNBP and the SFPP are also investigated.
{"title":"Fractional negative binomial and Pólya processes","authors":"P. Vellaisamy, A. Maheshwari","doi":"10.19195/0208-4147.38.1.5","DOIUrl":"https://doi.org/10.19195/0208-4147.38.1.5","url":null,"abstract":"In this paper, we define a fractional negative binomial process FNBP by replacing the Poisson process by a fractional Poisson process FPP in the gamma subordinated form of the negative binomial process. It is shown that the one-dimensional distributions of the FPP and the FNBP are not infinitely divisible. Also, the space fractional Pólya process SFPP is defined by replacing the rate parameter λ by a gamma random variable in the definition of the space fractional Poisson process. The properties of the FNBP and the SFPP and the connections to PDEs governing the density of the FNBP and the SFPP are also investigated.","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2013-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68000877","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2011-07-04DOI: 10.37190/0208-4147.40.1.8
R. Kapica, M. Ślȩczka
Markov chains arising from random iteration of functions $S_{theta}:Xto X$, $theta in Theta$, where $X$ is a Polish space and $Theta$ is arbitrary set of indices are considerd. At $xin X$, $theta$ is sampled from distribution $theta_x$ on $Theta$ and $theta_x$ are different for different $x$. Exponential convergence to a unique invariant measure is proved. This result is applied to case of random affine transformations on ${mathbb R}^d$ giving existence of exponentially attractive perpetuities with place dependent probabilities.
由函数$S_{theta}:Xto X$, $theta in Theta$随机迭代产生的马尔可夫链，其中$X$是波兰空间，$Theta$是任意索引集。在$xin X$上，$theta$是从$Theta$上的分布$theta_x$中采样的，$theta_x$对于不同的$x$是不同的。证明了指数收敛于唯一不变测度。该结果应用于${mathbb R}^d$上随机仿射变换的情况，给出了具有位置相关概率的指数吸引永续的存在性。
{"title":"Random iteration with place dependent probabilities","authors":"R. Kapica, M. Ślȩczka","doi":"10.37190/0208-4147.40.1.8","DOIUrl":"https://doi.org/10.37190/0208-4147.40.1.8","url":null,"abstract":"Markov chains arising from random iteration of functions $S_{theta}:Xto X$, $theta in Theta$, where $X$ is a Polish space and $Theta$ is arbitrary set of indices are considerd. At $xin X$, $theta$ is sampled from distribution $theta_x$ on $Theta$ and $theta_x$ are different for different $x$. Exponential convergence to a unique invariant measure is proved. This result is applied to case of random affine transformations on ${mathbb R}^d$ giving existence of exponentially attractive perpetuities with place dependent probabilities.","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2011-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69996895","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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