Pub Date : 2021-12-11DOI: 10.37863/tsp-9378222911-13
R. Zamini, S. Jomhoori
In recent years, in view of theory of empirical processes, authors have become more interested in the uniform analogue of the three fundamental theorems: the uniform law of large numbers of Glivenko-Cantelli type, the uniform central limit theorem for Donsker type and the functional law of the iterated logarithm (LIL). In this paper, under the bracketing entropy conditions, the uniform law of large numbers, uniform central limit theorem and the uniform LIL of Strassen type have been investigated in the case of length-biased and type I censoring.
{"title":"Uniform Limit Theorems under length-biased sampling and type I censoring","authors":"R. Zamini, S. Jomhoori","doi":"10.37863/tsp-9378222911-13","DOIUrl":"https://doi.org/10.37863/tsp-9378222911-13","url":null,"abstract":"\u0000In recent years, in view of theory of empirical processes, authors have become more interested in the uniform analogue of the three fundamental theorems: the uniform law of large numbers of Glivenko-Cantelli type, the uniform central limit theorem for Donsker type and the functional law of the iterated logarithm (LIL). In this paper, under the bracketing entropy conditions, the uniform law of large numbers, uniform central limit theorem and the uniform LIL of Strassen type have been investigated in the case of length-biased and type I censoring.\u0000","PeriodicalId":38143,"journal":{"name":"Theory of Stochastic Processes","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76337589","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-3616603423-59
I. Karnaukh
In this paper, we consider a path-wise sum of a Brownian motion plus a compound Poisson process with exponentially distributed positive and negative jumps with parameters that depend on some finite Markov chain. Using known fluctuation identities we investigate one-sided and two-sided exit problems generalizing some results for Kou's processes to the setting of regime switching models without exploiting the fluid embedding technique. The generating function for the hitting time of the state-dependent levels is analyzed. For the case of two states, the numerical examples are given.
{"title":"Exit problems for Kou's process in a Markovian environment","authors":"I. Karnaukh","doi":"10.37863/tsp-3616603423-59","DOIUrl":"https://doi.org/10.37863/tsp-3616603423-59","url":null,"abstract":"\u0000In this paper, we consider a path-wise sum of a Brownian motion plus a compound Poisson process\u0000with exponentially distributed positive and negative jumps with parameters that depend on some finite Markov chain. Using\u0000known fluctuation identities we investigate one-sided and two-sided\u0000exit problems generalizing some results for Kou's processes to the\u0000setting of regime switching models without exploiting the fluid embedding\u0000technique. The generating function for the hitting time of the state-dependent\u0000levels is analyzed. For the case of two states, the numerical examples\u0000are given.\u0000","PeriodicalId":38143,"journal":{"name":"Theory of Stochastic Processes","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80687431","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-5865169817-24
Fatma Ben Khadher, Y. Slaoui
In this research paper, we define a kernel estimator of the mode based on the recursive kernel density estimator developed by [23]. In addition, we establish its almost sure convergence under strong mixing hypothesis. Finally, we corroborate these theoretical results through numerical simulations.
{"title":"Strong consistency of the mode of multivariate recursive kernel density estimator under strong mixing hypothesis","authors":"Fatma Ben Khadher, Y. Slaoui","doi":"10.37863/tsp-5865169817-24","DOIUrl":"https://doi.org/10.37863/tsp-5865169817-24","url":null,"abstract":"\u0000In this research paper, we define a kernel estimator of the mode based on the recursive kernel density estimator developed by [23]. In addition, we establish its almost sure convergence under strong mixing hypothesis. Finally, we corroborate these theoretical results through numerical simulations.\u0000","PeriodicalId":38143,"journal":{"name":"Theory of Stochastic Processes","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89998306","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-4121179069-28
M. Belozerova
Two-dimensional stochastic differential equation with interaction is considered. The large time behavior of the distance between two solutions starting from different points is studied. A nonzero limit that characterize this distance together with the analogue of the triangle inequality for the map that characterize the limit distance are obtained.
{"title":"Asymptotic behavior of solutions to stochastic differential equations with interaction","authors":"M. Belozerova","doi":"10.37863/tsp-4121179069-28","DOIUrl":"https://doi.org/10.37863/tsp-4121179069-28","url":null,"abstract":"\u0000Two-dimensional stochastic differential equation with interaction is considered.\u0000The large time behavior of the distance between two solutions starting from different points is studied.\u0000A nonzero limit that characterize this distance together with the analogue of the triangle inequality for the map that characterize the limit distance are obtained.\u0000","PeriodicalId":38143,"journal":{"name":"Theory of Stochastic Processes","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83114476","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-0836718109-85
N. Zakharchenko, L.I. Nakonechna
The results on the existence of solutions for some discrete extremal problems with constraints were established. As an application the existence of a solution of a nonlinear eigenvalue problem was obtained.
建立了一类具有约束的离散极值问题解的存在性。作为应用，得到了一类非线性特征值问题解的存在性。
{"title":"On a discrete extremal problem with constraints","authors":"N. Zakharchenko, L.I. Nakonechna","doi":"10.37863/tsp-0836718109-85","DOIUrl":"https://doi.org/10.37863/tsp-0836718109-85","url":null,"abstract":"\u0000The results on the existence of solutions for some discrete extremal problems with constraints were established. As an application the existence of a solution of a nonlinear eigenvalue problem was obtained.\u0000","PeriodicalId":38143,"journal":{"name":"Theory of Stochastic Processes","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87934623","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 : 2021-06-18DOI: 10.37863/tsp-1140919749-78
A. Pilipenko, B. Povar
We consider a random walk Ŝ which has different increment distributions in positive and negative half-planes. In the upper half-plane the increments are mean-zero i.i.d. with finite variance. In the lower half-plane we consider two cases: increments are positive i.i.d. random variables with either a slowly varying tail or with a finite expectation. For the distributions with a slowly varying tails, we show that {Ŝ(nt)/√n} has no weak limit in D([0,1]); alternatively, the weak limit is a reflected Brownian motion.
{"title":"On a limit behaviour of a random walk penalised in the lower half-plane","authors":"A. Pilipenko, B. Povar","doi":"10.37863/tsp-1140919749-78","DOIUrl":"https://doi.org/10.37863/tsp-1140919749-78","url":null,"abstract":"\u0000We consider a random walk Ŝ which has different increment distributions in positive and negative half-planes.\u0000In the upper half-plane the increments are mean-zero i.i.d. with finite variance.\u0000In the lower half-plane we consider two cases: increments are positive i.i.d. random variables with either a slowly varying tail or with a finite expectation.\u0000For the distributions with a slowly varying tails, we show that {Ŝ(nt)/√n} has no weak limit in D([0,1]); alternatively, the weak limit is a reflected Brownian motion. \u0000","PeriodicalId":38143,"journal":{"name":"Theory of Stochastic Processes","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83312671","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 : 2021-03-08DOI: 10.37863/tsp-5223424922-78
Jasmina DJordjevi'c, A. Dorogovtsev
In this paper an analogue of the Clark-Ocone representation for solution to measure-valued equation with interaction is studied. It is proved that the integrand is absolutely continuous with respect to Lebesgue measure.
{"title":"Clark representation formula for the solution to equation with interaction","authors":"Jasmina DJordjevi'c, A. Dorogovtsev","doi":"10.37863/tsp-5223424922-78","DOIUrl":"https://doi.org/10.37863/tsp-5223424922-78","url":null,"abstract":"\u0000In this paper an analogue of the Clark-Ocone representation for solution to measure-valued equation with interaction is studied. It is proved that the integrand is absolutely continuous with respect to Lebesgue measure.\u0000","PeriodicalId":38143,"journal":{"name":"Theory of Stochastic Processes","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78197750","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 : 2021-02-22DOI: 10.37863/tsp-2295310746-81
V. Konarovskyi
We consider the system of sticky-reflected Brownian particles on the real line proposed in [4]. The model is a modification of the Howitt-Warren flow but now the diffusion rate of particles is inversely proportional to the mass which they transfer. It is known that the system consists of a finite number of distinct particles for almost all times. In this paper, we show that the system also admits an infinite number of distinct particles on a dense subset of the time interval if and only if the function responsible for the splitting of particles takes an infinite number of values.
{"title":"On number of particles in coalescing-fragmentating Wasserstein dynamics","authors":"V. Konarovskyi","doi":"10.37863/tsp-2295310746-81","DOIUrl":"https://doi.org/10.37863/tsp-2295310746-81","url":null,"abstract":"\u0000We consider the system of sticky-reflected Brownian particles on the real line proposed in [4]. The model is a modification of the Howitt-Warren flow but now the diffusion rate of particles is inversely proportional to the mass which they transfer. It is known that the system consists of a finite number of distinct particles for almost all times. In this paper, we show that the system also admits an infinite number of distinct particles on a dense subset of the time interval if and only if the function responsible for the splitting of particles takes an infinite number of values. \u0000","PeriodicalId":38143,"journal":{"name":"Theory of Stochastic Processes","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73256129","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 : 2020-12-21DOI: 10.37863/tsp-5986263728-06
V. Mandrekar, U. V. Naik-Nimbalkar
We prove the uniqueness of martingale solutions for stochastic partial differential equations generalizing the work in Mandrekar and Skorokhod (1998). The main idea used is to reduce this problem to the case in Mandrekar and Skorokhod using the techniques introduced in Filipović et al. (2010).
推广了Mandrekar和Skorokhod(1998)的工作，证明了随机偏微分方程鞅解的唯一性。使用的主要思想是使用filipoviki et al.(2010)中介绍的技术将这个问题减少到Mandrekar和Skorokhod的情况。
{"title":"Weak uniqueness of martingale solutions to stochastic partial differential equations in Hilbert spaces","authors":"V. Mandrekar, U. V. Naik-Nimbalkar","doi":"10.37863/tsp-5986263728-06","DOIUrl":"https://doi.org/10.37863/tsp-5986263728-06","url":null,"abstract":"\u0000We prove the uniqueness of martingale solutions for stochastic partial differential equations generalizing the work in Mandrekar and \u0000Skorokhod (1998). The main idea used is to reduce this problem to the case in Mandrekar and Skorokhod using the techniques introduced in \u0000Filipović et al. (2010).\u0000","PeriodicalId":38143,"journal":{"name":"Theory of Stochastic Processes","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72987669","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 : 2020-12-21DOI: 10.37863/tsp-7370403638-47
S. Bouzebda, M. Cherfi
In this paper, we extend the dual divergence approach to general semiparametric models and study dual divergence estimators for semiparametric models. Asymptotic properties such as consistency, asymptotic normality of the proposed estimators are deeply investigated by mean the sophisticated modern empirical theory. We investigate the exchangeably weighted estimators in this setting and establish the consistency. We finally consider the functional M-estimator and obtain its weak convergence result.
{"title":"General inference in semiparametric models through divergences and the duality technique with applications","authors":"S. Bouzebda, M. Cherfi","doi":"10.37863/tsp-7370403638-47","DOIUrl":"https://doi.org/10.37863/tsp-7370403638-47","url":null,"abstract":"\u0000In this paper, we extend the dual divergence approach to\u0000general semiparametric models and study dual divergence estimators for\u0000semiparametric models. Asymptotic properties such as consistency, asymptotic normality of the proposed estimators are deeply investigated by mean the sophisticated modern empirical theory. We investigate the exchangeably weighted estimators in this setting and establish the consistency. We finally consider the functional M-estimator and obtain its weak convergence result. \u0000","PeriodicalId":38143,"journal":{"name":"Theory of Stochastic Processes","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78612655","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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