Pub Date : 2024-07-18DOI: 10.3842/tsp-2702069172-88
N.N. Ganikhodjaev
We consider a stochastic process generated by 1-D Ising model with competing interactions and describe all distributions of this process. It is shown that the set of all limit Gibbs measures, i.e. phase diagram, consist of ferromagnetic, anti-ferromagnetic, paramagnetic and modulated phases. Also it is proven that on the set of ferromagnetic phases one can reach the phase transition.
{"title":"Stochastic process generated by 1-D Ising model with competing interactions","authors":"N.N. Ganikhodjaev","doi":"10.3842/tsp-2702069172-88","DOIUrl":"https://doi.org/10.3842/tsp-2702069172-88","url":null,"abstract":"\u0000We consider a stochastic process generated by 1-D Ising model with competing interactions and describe all distributions of this process.\u0000It is shown that the set of all limit Gibbs measures, i.e. phase diagram, consist of ferromagnetic, anti-ferromagnetic, paramagnetic and modulated phases.\u0000Also it is proven that on the set of ferromagnetic phases one can reach the phase transition. \u0000","PeriodicalId":38143,"journal":{"name":"Theory of Stochastic Processes","volume":" 64","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141825136","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}
Pub Date : 2024-07-18DOI: 10.3842/tsp-0731915872-49
S. Khardani, W. Nefzi, C. Thabet
In this paper, we address the case of a randomly right-censored model when the data exhibit some kind of dependency. We build and study a new nonparametric regression estimator by using the mean squared relative error as a loss function. Under classical conditions, we establish the uniform consistency with rate and asymptotic normality of the estimator suitably normalized.
{"title":"Relative error prediction from censored data under α-mixing condition","authors":"S. Khardani, W. Nefzi, C. Thabet","doi":"10.3842/tsp-0731915872-49","DOIUrl":"https://doi.org/10.3842/tsp-0731915872-49","url":null,"abstract":"\u0000In this paper, we address the case of a randomly right-censored model when the data exhibit some kind of dependency. We build and study a new nonparametric regression estimator by using the mean squared relative error as a loss function. Under classical conditions, we establish the uniform consistency with rate and asymptotic normality of the estimator suitably normalized. \u0000","PeriodicalId":38143,"journal":{"name":"Theory of Stochastic Processes","volume":" 2","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141826571","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}
Pub Date : 2024-07-18DOI: 10.3842/tsp-1833768554-46
G. Cleanthous
We consider multivariate isotropic random fields on the ball Bd. We first study their regularity properties in terms of Sobolev spaces. We further derive conditions guaranteeing the Hölder continuity of their covariance kernels and we prove the existence of sample Hölder continuous modifications for Gaussian random fields. Furthermore, we measure the error of truncated approximations of the corresponding series' representations. Moreover our developments are supported by numerical experiments. The majority of our results are new for multivariate random fields indexed over other domains, too. We express some of them for the case of the sphere.
{"title":"On the properties of multivariate isotropic Random fields on the Ball","authors":"G. Cleanthous","doi":"10.3842/tsp-1833768554-46","DOIUrl":"https://doi.org/10.3842/tsp-1833768554-46","url":null,"abstract":"\u0000We consider multivariate isotropic random fields on the ball Bd.\u0000We first study their regularity properties in terms of Sobolev spaces.\u0000We further derive conditions guaranteeing the Hölder continuity of their covariance kernels and we prove the existence of sample Hölder continuous modifications for Gaussian random fields.\u0000Furthermore, we measure the error of truncated approximations of the corresponding series' representations.\u0000Moreover our developments are supported by numerical experiments.\u0000The majority of our results are new for multivariate random fields indexed over other domains, too.\u0000We express some of them for the case of the sphere.\u0000","PeriodicalId":38143,"journal":{"name":"Theory of Stochastic Processes","volume":" 26","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141827612","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}
Pub Date : 2022-12-27DOI: 10.37863/tsp-2899660400-77
Y. Slaoui, A. Jmaei
If a regression function has a bounded support, the kernel estimates often exceed the boundaries and are therefore biased on and near these limits. In this paper, we focus on mitigating this boundary problem. We apply Bernstein polynomials and the Robbins-Monro algorithm to construct a non-recursive and recursive regression estimator. We study the asymptotic properties of these estimators, and we compare them with those of the Nadaraya-Watson estimator and the generalized Révész estimator introduced by [21]. In addition, through some simulation studies, we show that our non-recursive estimator has the lowest integrated root mean square error (ISE) in most of the considered cases. Finally, using a set of real data, we demonstrate how our non-recursive and recursive regression estimators can lead to very satisfactory estimates, especially near the boundaries.
{"title":"Recursive and non-recursive regression estimators using Bernstein polynomials","authors":"Y. Slaoui, A. Jmaei","doi":"10.37863/tsp-2899660400-77","DOIUrl":"https://doi.org/10.37863/tsp-2899660400-77","url":null,"abstract":"\u0000If a regression function has a bounded support, the kernel estimates often exceed the boundaries and are therefore biased on and near these limits. In this paper, we focus on mitigating this boundary problem. We apply Bernstein polynomials and the Robbins-Monro algorithm to construct a non-recursive and recursive regression estimator. We study the asymptotic properties of these estimators, and we compare them with those of the Nadaraya-Watson estimator and the generalized Révész estimator introduced by [21]. In addition, through some simulation studies, we show that our non-recursive estimator has the lowest integrated root mean square error (ISE) in most of the considered cases. Finally, using a set of real data, we demonstrate how our non-recursive and recursive regression estimators can lead to very satisfactory estimates, especially near the boundaries.\u0000","PeriodicalId":38143,"journal":{"name":"Theory of Stochastic Processes","volume":"1992 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88996881","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}
Pub Date : 2022-12-27DOI: 10.37863/tsp-3639580555-63
Y. Khusanbaev, Kh.A. Toshkulov
In this paper we consider critical Galton-Watson branching processes Zk, k ≥ 0 in the case when the number of direct offspring of one particle has infinite variance. Limit theorems for conditional distributions of Zk are proved.
{"title":"Limit theorems for conditional distributions of critical Galton-Watson branching processes without finite variance","authors":"Y. Khusanbaev, Kh.A. Toshkulov","doi":"10.37863/tsp-3639580555-63","DOIUrl":"https://doi.org/10.37863/tsp-3639580555-63","url":null,"abstract":"\u0000In this paper we consider critical Galton-Watson branching processes Zk, k ≥ 0 in the case when the number of direct offspring of one particle has infinite variance. Limit theorems for conditional distributions of Zk are proved.\u0000","PeriodicalId":38143,"journal":{"name":"Theory of Stochastic Processes","volume":"31 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90517391","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}
Pub Date : 2022-12-27DOI: 10.37863/tsp-7754833420-58
S. Slama, Y. Slaoui, H. Fathallah
In this research paper, we elaborate an extension of the semi-recursive kernel-type regression function estimator. We investigate the asymptotic properties of this estimator and compare them with non-recursive Nadaraya Watson regression estimator. From this perspective, we first calculate the bias and the variance of the proposed estimator which strongly depend on the choice of three parameters, namely the stepsizes (βn) and (γn) as well as the bandwidth (hn) chosen using one of the best methods of bandwidth selection, the bootstrap approach compared to the plug-in method. An appropriate choice of those parameters yields that, under some conditions, the MSE (Mean Squared Error) of the proposed estimator can be smaller than that of Nadaraya Watson's estimator. We corroborate our theoretical results through simulations studies and by considering two real dataset applications, the French Hospital Data of COVID-19 epidemic as well as the Plasmodium Falciparum Parasite Load (PL).
{"title":"The stochastic approximation method for semi-recursive multivariate kernel-type regression estimation","authors":"S. Slama, Y. Slaoui, H. Fathallah","doi":"10.37863/tsp-7754833420-58","DOIUrl":"https://doi.org/10.37863/tsp-7754833420-58","url":null,"abstract":"\u0000In this research paper, we elaborate an extension of the semi-recursive kernel-type regression function estimator. We investigate the asymptotic properties of this estimator and compare them with non-recursive Nadaraya Watson regression estimator. From this perspective, we first calculate the bias and the variance of the proposed estimator which strongly depend on the choice of three parameters, namely the stepsizes (βn) and (γn) as well as the bandwidth (hn) chosen using one of the best methods of bandwidth selection, the bootstrap approach compared to the plug-in method. An appropriate choice of those parameters yields that, under some conditions, the MSE (Mean Squared Error) of the proposed estimator can be smaller than that of Nadaraya Watson's estimator. \u0000 We corroborate our theoretical results through simulations studies and by considering two real dataset applications, the French Hospital Data of COVID-19 epidemic as well as the Plasmodium Falciparum Parasite Load (PL).\u0000","PeriodicalId":38143,"journal":{"name":"Theory of Stochastic Processes","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75946595","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}
Pub Date : 2022-12-27DOI: 10.37863/tsp-3130168706-50
M. El Jamali
In this paper, we study the generalized backward stochastic differential equations driven by inhomogeneous Lévy processes (GBSDELs in short). We establish the existence and uniqueness of solution by using Picard's iteration setting under non-deterministic Lipschitz and monotone condition.
{"title":"Generalized BSDEs for time inhomogeneous Lévy processes under non-deterministic Lipschitz coefficient","authors":"M. El Jamali","doi":"10.37863/tsp-3130168706-50","DOIUrl":"https://doi.org/10.37863/tsp-3130168706-50","url":null,"abstract":"\u0000In this paper, we study the generalized backward stochastic differential equations driven by inhomogeneous Lévy processes (GBSDELs in short).\u0000We establish the existence and uniqueness of solution by using Picard's iteration setting under non-deterministic Lipschitz and monotone condition.\u0000","PeriodicalId":38143,"journal":{"name":"Theory of Stochastic Processes","volume":"46 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75235756","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}
Pub Date : 2022-12-27DOI: 10.37863/tsp-0919442573-40
Y. Miniailyk
In this article we consider the generalization of Ehrenfest model, where at each moment of time not 1 but some k of n particles go from one box to another. We describe this process by a sequence of Bernoulli random vectors. We define related Bernoulli noise on a set of continuous functions for different times, and prove that it converges to Ornstein-Uhlenbeck sequence of Gaussian white noises when number of particles tends to infinity.
{"title":"Gaussian noise related to generalised Ehrenfest model","authors":"Y. Miniailyk","doi":"10.37863/tsp-0919442573-40","DOIUrl":"https://doi.org/10.37863/tsp-0919442573-40","url":null,"abstract":"\u0000In this article we consider the generalization of Ehrenfest model, where at each moment of time not 1 but some k of n particles go from one box to another. We describe this process by a sequence of Bernoulli random vectors. We define related Bernoulli noise on a set of continuous functions for different times, and prove that it converges to Ornstein-Uhlenbeck sequence of Gaussian white noises when number of particles tends to infinity.\u0000","PeriodicalId":38143,"journal":{"name":"Theory of Stochastic Processes","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87300965","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}
Pub Date : 2021-12-11DOI: 10.37863/tsp-1348277559-92
A. Dorogovtsev, I. Nishchenko
A new class of loop-erased random walks (LERW) on a finite set, defined as functionals from a Markov chain is presented. We propose a scheme in which, in contrast to the general settings of LERW, the loop-erasure is performed on a non-markovian sequence and moreover, not all loops are erased with necessity. We start with a special example of a random walk with loops, the number of which at every moment of time does not exceed a given fixed number. Further we consider loop-erased random walks, for which loops are erased at random moments of time that are hitting times for a Markov chain. The asymptotics of the normalized length of such loop-erased walks is established. We estimate also the speed of convergence of the normalized length of the loop-erased random walk on a finite group to the Rayleigh distribution.
{"title":"Loop-erased random walks associated with Markov processes","authors":"A. Dorogovtsev, I. Nishchenko","doi":"10.37863/tsp-1348277559-92","DOIUrl":"https://doi.org/10.37863/tsp-1348277559-92","url":null,"abstract":"\u0000A new class of loop-erased random walks (LERW) on a finite set, defined as functionals from a Markov chain is presented.\u0000We propose a scheme in which, in contrast to the general settings of LERW, the loop-erasure is performed on a non-markovian sequence and moreover, not all loops are erased with necessity. We start with a special example of a random walk with loops, the number of which at every moment of time does not exceed a given fixed number. Further we consider loop-erased random walks, for which loops are erased at random moments of time that are hitting times for a Markov chain. \u0000The asymptotics of the normalized length of such loop-erased walks is established. \u0000We estimate also the speed of convergence of the normalized length of the loop-erased random walk on a finite group to the Rayleigh distribution. \u0000","PeriodicalId":38143,"journal":{"name":"Theory of Stochastic Processes","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75503189","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}
Pub Date : 2021-12-11DOI: 10.37863/tsp-9489058429-71
V. Kanišauskas, K. Kanišauskienė
We consider first and second error probabilities of asymptotically optimal tests (Neyman-Pearson, minimax, Bayesian) when two simple hypotheses H1t and H2t parametrized by time t ≥ 0 are tested under the observation Xt of arbitrary nature. The paper provides details on the conditions of asymptotic decrease of probabilities of optimal criteria errors determined by α type Hellinger integral between measures P1t and P2t, demonstrating that in the case of minimax and Bayesian criteria it is sufficient to investigate Hellinger integral, when α ∈ (0,1), and in the case of Neyman-Pearson criterion it is observed only in the environment of point α=1. Whereas Kullback-Leibler information distance is always larger than Chernoff distance; we discover that, in the case of Neyman-Pearson criterion, the probability of type II error decreases faster than in the cases of minimax or Bayesian criteria. This is proven by the examples of marked point processes of the i.i.d. case, non-homogeneous Poisson process and the geometric renewal process presented at the end of the paper.
{"title":"Asymptotics of error probabilities of optimal tests","authors":"V. Kanišauskas, K. Kanišauskienė","doi":"10.37863/tsp-9489058429-71","DOIUrl":"https://doi.org/10.37863/tsp-9489058429-71","url":null,"abstract":"\u0000We consider first and second error probabilities of asymptotically optimal tests (Neyman-Pearson, minimax, Bayesian) when two simple hypotheses H1t and H2t parametrized by time t ≥ 0 are tested under the observation Xt of arbitrary nature. \u0000The paper provides details on the conditions of asymptotic decrease of probabilities of optimal criteria errors determined by α type Hellinger integral between measures P1t and P2t, demonstrating that in the case of minimax and Bayesian criteria it is sufficient to investigate Hellinger integral, when α ∈ (0,1), and in the case of Neyman-Pearson criterion it is observed only in the environment of point α=1.\u0000Whereas Kullback-Leibler information distance is always larger than Chernoff distance; we discover that, in the case of Neyman-Pearson criterion, the probability of type II error decreases faster than in the cases of minimax or Bayesian criteria. This is proven by the examples of marked point processes of the i.i.d. case, non-homogeneous Poisson process and the geometric renewal process presented at the end of the paper.\u0000","PeriodicalId":38143,"journal":{"name":"Theory of Stochastic Processes","volume":"55 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76195944","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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