Pub Date : 2021-07-11DOI: 10.37190/0208-4147.00044
M. Vidmar
Continuous-state branching processes (CSBPs) with immigration (CBIs), stopped on hitting zero, are generalized by allowing the process governing immigration to be any L'evy process without negative jumps. Unlike the CBIs, these newly introduced processes do not appear to satisfy any natural affine property on the level of the Laplace transforms of the semigroups. Basic properties are noted. Explicit formulae (on neighborhoods of infinity) for the Laplace transforms of the first passage times downwards and of the explosion time are derived.
{"title":"Continuous-state branching processes with spectrally positive migration","authors":"M. Vidmar","doi":"10.37190/0208-4147.00044","DOIUrl":"https://doi.org/10.37190/0208-4147.00044","url":null,"abstract":"Continuous-state branching processes (CSBPs) with immigration (CBIs), stopped on hitting zero, are generalized by allowing the process governing immigration to be any L'evy process without negative jumps. Unlike the CBIs, these newly introduced processes do not appear to satisfy any natural affine property on the level of the Laplace transforms of the semigroups. Basic properties are noted. Explicit formulae (on neighborhoods of infinity) for the Laplace transforms of the first passage times downwards and of the explosion time are derived.","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2021-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48301094","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 : 2021-02-17DOI: 10.37190/0208-4147.00045
David Berger, Franziska Kuhn, R. Schilling
We give new proofs of certain equivalent conditions for the existence of generalized moments of a L'evy process $(X_t)_{tgeq 0}$; in particular, the existence of a generalized $g$-moment is equivalent to the uniform integrability of $(g(X_t))_{tin [0,1]}$. As a consequence, certain functions of a L'evy process which are integrable and local martingales are already true martingales. Our methods extend to moments of stochastically continuous additive processes, and we give new, short proofs for the characterization of lattice distributions and the transience of L'evy processes.
{"title":"Lévy Processes, Generalized Moments and Uniform Integrability","authors":"David Berger, Franziska Kuhn, R. Schilling","doi":"10.37190/0208-4147.00045","DOIUrl":"https://doi.org/10.37190/0208-4147.00045","url":null,"abstract":"We give new proofs of certain equivalent conditions for the existence of generalized moments of a L'evy process $(X_t)_{tgeq 0}$; in particular, the existence of a generalized $g$-moment is equivalent to the uniform integrability of $(g(X_t))_{tin [0,1]}$. As a consequence, certain functions of a L'evy process which are integrable and local martingales are already true martingales. Our methods extend to moments of stochastically continuous additive processes, and we give new, short proofs for the characterization of lattice distributions and the transience of L'evy processes.","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2021-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46232627","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 : 2020-07-11DOI: 10.37190/0208-4147.41.2.6
B. Boufoussi, Yassine Nachit
Our aim in this paper is to improve Holder continuity results for the bifractional Brownian motion (bBm) $(B^{alpha,beta}(t))_{tin[0,1] }$ with $0 frac{1}{2}$ in the Holder spaces $mathcal{C}^{gamma}$, with $gamma
本文的目的是改进Holder空间$mathcal{C}^{gamma}$和$gamma
{"title":"On the Besov regularity of the bifractional Brownian motion","authors":"B. Boufoussi, Yassine Nachit","doi":"10.37190/0208-4147.41.2.6","DOIUrl":"https://doi.org/10.37190/0208-4147.41.2.6","url":null,"abstract":"Our aim in this paper is to improve Holder continuity results for the bifractional Brownian motion (bBm) $(B^{alpha,beta}(t))_{tin[0,1] }$ with $0 frac{1}{2}$ in the Holder spaces $mathcal{C}^{gamma}$, with $gamma<alpha beta$.","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2020-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45844022","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 : 2020-06-25DOI: 10.37190/0208-4147.00056
Amit Kumar, P. Vellaisamy, F. Viens
In this article, we obtain, for the total variance distance, the error bounds between Poisson and convolution of power series distributions via Stein's method. This provides a unified approach to many known discrete distributions. Several Poisson limit theorems follow as corollaries from our bounds. As applications, we compare the Poisson approximation results with the negative binomial approximation results, for the sums of Bernoulli, geometric, and logarithmic series random variables.
{"title":"Poisson Approximation to the Convolution of Power Series Distributions","authors":"Amit Kumar, P. Vellaisamy, F. Viens","doi":"10.37190/0208-4147.00056","DOIUrl":"https://doi.org/10.37190/0208-4147.00056","url":null,"abstract":"In this article, we obtain, for the total variance distance, the error bounds between Poisson and convolution of power series distributions via Stein's method. This provides a unified approach to many known discrete distributions. Several Poisson limit theorems follow as corollaries from our bounds. As applications, we compare the Poisson approximation results with the negative binomial approximation results, for the sums of Bernoulli, geometric, and logarithmic series random variables.","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2020-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42751020","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 : 2020-06-11DOI: 10.37190/0208-4147.00020
Gerardo Barrera Vargas
In this article, we consider the Langevin dynamics on $mathbb{R}^d$ with an overdamped vector field and driven by Brownian motion of small amplitude $sqrt{epsilon}$, $epsilon>0$. Under suitable conditions on the vector field, it is well-known that it possesses a unique invariant probability measure $mu^{epsilon}$. As $epsilon$ tends to zero, we prove that the probability measure $epsilon^{d/2} mu^{epsilon}(sqrt{epsilon}mathrm{d}x)$ converges in the $2$-Wasserstein distance to a Gaussian measure with zero-mean vector and non-degenerate covariance matrix which solves a Lyapunov matrix equation. We emphasize that generically no explicit formula for $mu^{epsilon}$ can be found.
{"title":"Limit behavior of the invariant measure for Langevin dynamics","authors":"Gerardo Barrera Vargas","doi":"10.37190/0208-4147.00020","DOIUrl":"https://doi.org/10.37190/0208-4147.00020","url":null,"abstract":"In this article, we consider the Langevin dynamics on $mathbb{R}^d$ with an overdamped vector field and driven by Brownian motion of small amplitude $sqrt{epsilon}$, $epsilon>0$. Under suitable conditions on the vector field, it is well-known that it possesses a unique invariant probability measure $mu^{epsilon}$. As $epsilon$ tends to zero, we prove that the probability measure $epsilon^{d/2} mu^{epsilon}(sqrt{epsilon}mathrm{d}x)$ converges in the $2$-Wasserstein distance to a Gaussian measure with zero-mean vector and non-degenerate covariance matrix which solves a Lyapunov matrix equation. We emphasize that generically no explicit formula for $mu^{epsilon}$ can be found.","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2020-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69996728","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 : 2020-06-04DOI: 10.37190/0208-4147.41.2.2
Qianqian Zhou, A. Sakhanenko, Junyi Guo
Lundberg-type inequalities for ruin probabilities of non-homogeneous risk models are presented in this paper. By employing martingale method, the upper bounds of ruin probabilities are obtained for the general risk models under weak assumptions. In addition, several risk models, including the newly defined united risk model and quasi-periodic risk model with interest rate, are studied.
{"title":"Exponential bounds of ruin probabilities for non-homogeneous risk models","authors":"Qianqian Zhou, A. Sakhanenko, Junyi Guo","doi":"10.37190/0208-4147.41.2.2","DOIUrl":"https://doi.org/10.37190/0208-4147.41.2.2","url":null,"abstract":"Lundberg-type inequalities for ruin probabilities of non-homogeneous risk models are presented in this paper. By employing martingale method, the upper bounds of ruin probabilities are obtained for the general risk models under weak assumptions. In addition, several risk models, including the newly defined united risk model and quasi-periodic risk model with interest rate, are studied.","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2020-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41936505","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 : 2020-01-01DOI: 10.37190/0208-4147.41.2.4
A. Adler, P. Matuła
We examine order statistics from a two sided Pareto distribution. It turns out that the smallest two order statistics and the two largest order statistics have very unusual limits. We obtain strong and weak exact laws for the smallest and the largest order statistics. For such statistics we also study the generalized law of the iterated logarithm. For the second smallest and second largest order statistics we prove the central limit theorem even though their second moment is infinite. 2010 AMS Mathematics Subject Classification: Primary: 60F05; Secondary: 60F15.
{"title":"Unusual limit theorems for the two tailed Pareto distribution","authors":"A. Adler, P. Matuła","doi":"10.37190/0208-4147.41.2.4","DOIUrl":"https://doi.org/10.37190/0208-4147.41.2.4","url":null,"abstract":"We examine order statistics from a two sided Pareto distribution. It turns out that the smallest two order statistics and the two largest order statistics have very unusual limits. We obtain strong and weak exact laws for the smallest and the largest order statistics. For such statistics we also study the generalized law of the iterated logarithm. For the second smallest and second largest order statistics we prove the central limit theorem even though their second moment is infinite. 2010 AMS Mathematics Subject Classification: Primary: 60F05; Secondary: 60F15.","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69997131","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 : 2020-01-01DOI: 10.37190/0208-4147.41.1.4
M. Sanatgar, A. Dolati, M. Amini
We propose a class of weighted rank correlation measures extending Spearman’s rho. This class consists of two types of measures. The first type that extends Blest’s rank correlation, places more emphasis on the agreement on top ranks. The second one places more emphasis on the agreement on the bottom ranks. The asymptotic distribution of the proposed measures and some of their properties are studied. A simulation study is performed to compare the performance of the proposed statistics for testing independence by using asymptotic relative efficiency calculations. 2010 AMS Mathematics Subject Classification: Primary: 01A23; Secondary: 45B67.
{"title":"A class of weighted rank correlation measures","authors":"M. Sanatgar, A. Dolati, M. Amini","doi":"10.37190/0208-4147.41.1.4","DOIUrl":"https://doi.org/10.37190/0208-4147.41.1.4","url":null,"abstract":"We propose a class of weighted rank correlation measures extending Spearman’s rho. This class consists of two types of measures. The first type that extends Blest’s rank correlation, places more emphasis on the agreement on top ranks. The second one places more emphasis on the agreement on the bottom ranks. The asymptotic distribution of the proposed measures and some of their properties are studied. A simulation study is performed to compare the performance of the proposed statistics for testing independence by using asymptotic relative efficiency calculations. 2010 AMS Mathematics Subject Classification: Primary: 01A23; Secondary: 45B67.","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69996960","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 : 2020-01-01DOI: 10.37190/0208-4147.41.2.10
Chi Yao, Rui Wang, Ling Chen, Xuejun Wang
{"title":"Complete f-moment convergence of moving average process and its application to nonparametric regression models","authors":"Chi Yao, Rui Wang, Ling Chen, Xuejun Wang","doi":"10.37190/0208-4147.41.2.10","DOIUrl":"https://doi.org/10.37190/0208-4147.41.2.10","url":null,"abstract":"","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":"107 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69997108","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 : 2020-01-01DOI: 10.37190/0208-4147.41.1.9
T. Byczkowski, T. Żak
{"title":"Borell and Landau-Shepp inequalities for Cauchy-type measures","authors":"T. Byczkowski, T. Żak","doi":"10.37190/0208-4147.41.1.9","DOIUrl":"https://doi.org/10.37190/0208-4147.41.1.9","url":null,"abstract":"","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69997072","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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