Pub Date : 2019-06-10DOI: 10.19195/0208-4147.39.1.12
Lahcen Oussi, J. Wysoczánski
Analogues of the classical Central Limit Theorem are proved in the noncommutative setting of random variables which are bmindependent and indexed by elements of positive non-symmetric cones, such as the circular cone, sectors in Euclidean spaces and the Vinberg cone. The geometry of the cones is shown to play a crucial role and the related volume characteristics of the cones is shown.
{"title":"bm-Central Limit Theorems associated with non-symmetric positive cones","authors":"Lahcen Oussi, J. Wysoczánski","doi":"10.19195/0208-4147.39.1.12","DOIUrl":"https://doi.org/10.19195/0208-4147.39.1.12","url":null,"abstract":"Analogues of the classical Central Limit Theorem are proved in the noncommutative setting of random variables which are bmindependent and indexed by elements of positive non-symmetric cones, such as the circular cone, sectors in Euclidean spaces and the Vinberg cone. The geometry of the cones is shown to play a crucial role and the related volume characteristics of the cones is shown.","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47239316","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 : 2019-06-10DOI: 10.19195/0208-4147.39.1.5
J. Tugaut
The current work deals with the granular media equation whose probabilistic interpretation is the McKean–Vlasov diffusion. It is well known that the Laplacian provides a regularization of the solution. Indeed, for any t > 0, the solution is absolutely continuous with respect to the Lebesgue measure. It has also been proved that all the moments are bounded for positive t. However, the finiteness of the entropy of the solution is a new result which will be presented here.
{"title":"Finiteness of entropy for granular media equations","authors":"J. Tugaut","doi":"10.19195/0208-4147.39.1.5","DOIUrl":"https://doi.org/10.19195/0208-4147.39.1.5","url":null,"abstract":"The current work deals with the granular media equation whose probabilistic interpretation is the McKean–Vlasov diffusion. It is well known that the Laplacian provides a regularization of the solution. Indeed, for any t > 0, the solution is absolutely continuous with respect to the Lebesgue measure. It has also been proved that all the moments are bounded for positive t. However, the finiteness of the entropy of the solution is a new result which will be presented here.","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43023678","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 : 2019-06-10DOI: 10.19195/0208-4147.39.1.2
Shuhua Chang, Deli Li, A. Rosalsky
Let 0 < p ≤ 2, let {Xn; n ≥ 1} be a sequence of independent copies of a real-valued random variable X, and set Sn = X1 + . . . + Xn, n ≥ 1. Motivated by a theorem of Mikosch 1984, this note is devoted to establishing a strong law of large numbers for the sequence {max1≤k≤n |Sk| ; n ≥ 1}. More specifically, necessary and sufficient conditions are given forlimn→∞ max1≤k≤n |Sk|log n−1 = e1/p a.s.,where log x = loge max{e, x}, x ≥ 0.
{"title":"Strong laws of large numbers for the sequence of the maximum of partial sums of i.i.d. random variables","authors":"Shuhua Chang, Deli Li, A. Rosalsky","doi":"10.19195/0208-4147.39.1.2","DOIUrl":"https://doi.org/10.19195/0208-4147.39.1.2","url":null,"abstract":"Let 0 < p ≤ 2, let {Xn; n ≥ 1} be a sequence of independent copies of a real-valued random variable X, and set Sn = X1 + . . . + Xn, n ≥ 1. Motivated by a theorem of Mikosch 1984, this note is devoted to establishing a strong law of large numbers for the sequence {max1≤k≤n |Sk| ; n ≥ 1}. More specifically, necessary and sufficient conditions are given forlimn→∞ max1≤k≤n |Sk|log n−1 = e1/p a.s.,where log x = loge max{e, x}, x ≥ 0.","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41801005","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 : 2019-06-10DOI: 10.19195/0208-4147.39.1.6
A. Arefi, R. Pourtaheri
In this paper, we introduce a technique to produce a new family of tempered stable distributions. We call this family asymmetrically tempered stable distributions.We provide two examples of this family named asymmetrically classical modified tempered stable ACMTS and asymmetrically modified classical tempered stable AMCTS distributions. Since the tempered stable distributions are infinitely divisible, Levy processes can be induced by the ACMTS and AMCTS distributions. The properties of these distributions will be discussed along with the advantages in applying them to financial modeling. Furthermore, we develop exponential Levy models for them. To demonstrate the advantages of the exponential Levy ACMTS and AMCTS models, we estimate parameters for the S&P 500 Index.
{"title":"Asymmetrically tempered stable distributions with applications to finance","authors":"A. Arefi, R. Pourtaheri","doi":"10.19195/0208-4147.39.1.6","DOIUrl":"https://doi.org/10.19195/0208-4147.39.1.6","url":null,"abstract":"In this paper, we introduce a technique to produce a new family of tempered stable distributions. We call this family asymmetrically tempered stable distributions.We provide two examples of this family named asymmetrically classical modified tempered stable ACMTS and asymmetrically modified classical tempered stable AMCTS distributions. Since the tempered stable distributions are infinitely divisible, Levy processes can be induced by the ACMTS and AMCTS distributions. The properties of these distributions will be discussed along with the advantages in applying them to financial modeling. Furthermore, we develop exponential Levy models for them. To demonstrate the advantages of the exponential Levy ACMTS and AMCTS models, we estimate parameters for the S&P 500 Index.","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43738991","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 : 2019-06-10DOI: 10.19195/0208-4147.39.1.4
A. Imomov
We investigate limit properties of discrete time branching processes with application of the theory of regularly varying functions in the sense of Karamata. In the critical situation we suppose that the offspring probability generating function has an infinite second moment but its tail regularly varies. In the noncritical case, the finite moment of type Ε[x ln x] is required. The lemma on the asymptotic representation of the generating function of the process and its differential analogue will underlie our conclusions.
{"title":"On a limit structure of the Galton–Watson branching processes with regularly varying generating functions","authors":"A. Imomov","doi":"10.19195/0208-4147.39.1.4","DOIUrl":"https://doi.org/10.19195/0208-4147.39.1.4","url":null,"abstract":"We investigate limit properties of discrete time branching processes with application of the theory of regularly varying functions in the sense of Karamata. In the critical situation we suppose that the offspring probability generating function has an infinite second moment but its tail regularly varies. In the noncritical case, the finite moment of type Ε[x ln x] is required. The lemma on the asymptotic representation of the generating function of the process and its differential analogue will underlie our conclusions.","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49614751","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 : 2019-06-10DOI: 10.19195/0208-4147.39.1.7
Mohamed Ait Ouahra, Raby Guerbaz, H. Ouahhabi, A. Sghir
In this paper, by using a Fourier analytic approach, we investigate sample path properties of the fractional derivatives of multifractional Brownian motion local times. We also show that those additive functionals satisfy a property of local asymptotic self-similarity. As a consequence, we derive some local limit theorems for the occupation time of multifractional Brownian motion in the space of continuous functions.
{"title":"Occupation time problem for multifractional Brownian motion","authors":"Mohamed Ait Ouahra, Raby Guerbaz, H. Ouahhabi, A. Sghir","doi":"10.19195/0208-4147.39.1.7","DOIUrl":"https://doi.org/10.19195/0208-4147.39.1.7","url":null,"abstract":"In this paper, by using a Fourier analytic approach, we investigate sample path properties of the fractional derivatives of multifractional Brownian motion local times. We also show that those additive functionals satisfy a property of local asymptotic self-similarity. As a consequence, we derive some local limit theorems for the occupation time of multifractional Brownian motion in the space of continuous functions. ","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42105380","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 : 2019-06-10DOI: 10.19195/0208-4147.39.1.8
T. Inglot
New data driven score tests for testing goodness of fit of the Poisson distribution are proposed. They are direct applications of the general construction of data driven goodness-of-fit tests for composite hypotheses developed in Inglot et al. 1997. By a simulation study it is shown that these tests perform almost equally well as the best known solutions for standard alternatives and outperform them for more difficult alternatives.
{"title":"Data driven efficient score tests for Poissonity","authors":"T. Inglot","doi":"10.19195/0208-4147.39.1.8","DOIUrl":"https://doi.org/10.19195/0208-4147.39.1.8","url":null,"abstract":"New data driven score tests for testing goodness of fit of the Poisson distribution are proposed. They are direct applications of the general construction of data driven goodness-of-fit tests for composite hypotheses developed in Inglot et al. 1997. By a simulation study it is shown that these tests perform almost equally well as the best known solutions for standard alternatives and outperform them for more difficult alternatives.","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47418969","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 : 2019-06-10DOI: 10.19195/0208-4147.39.1.9
Chenlu Zhuansun, Xiaoxin Li
In this paper, we will study the recursive density estimators of the probability density function for widely orthant dependent WOD random variables. The complete consistency and complete convergence rate are established under some general conditions.
{"title":"Complete consistency for recursive probability density estimator of widely orthant dependent samples","authors":"Chenlu Zhuansun, Xiaoxin Li","doi":"10.19195/0208-4147.39.1.9","DOIUrl":"https://doi.org/10.19195/0208-4147.39.1.9","url":null,"abstract":"In this paper, we will study the recursive density estimators of the probability density function for widely orthant dependent WOD random variables. The complete consistency and complete convergence rate are established under some general conditions.","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49506739","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 : 2019-06-10DOI: 10.19195/0208-4147.39.1.11
B. Pacchiarotti
We study the asymptotic behavior of a Gaussian process conditioned to n linear functionals of its paths and of the bridge of such a process. In particular, functional large deviation results are stated for small time. Two examples are considered.
{"title":"Large deviations for generalized conditioned Gaussian processes and their bridges","authors":"B. Pacchiarotti","doi":"10.19195/0208-4147.39.1.11","DOIUrl":"https://doi.org/10.19195/0208-4147.39.1.11","url":null,"abstract":"We study the asymptotic behavior of a Gaussian process conditioned to n linear functionals of its paths and of the bridge of such a process. In particular, functional large deviation results are stated for small time. Two examples are considered.","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45839002","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 : 2019-01-10DOI: 10.37190/0208-4147.41.2.8
Idir Arab, M. Hadjikyriakou, P. E. Oliveira
In this paper we prove that two heterogeneous parallel systems with independent exponentially distributed components are comparable via the star transform order while the comparison via the convex transform fails. The latter conclusion provides a partial answer to a problem that remained open for a decade.
{"title":"On a conjecture about the comparability of parallel systems with respect to the convex transform order","authors":"Idir Arab, M. Hadjikyriakou, P. E. Oliveira","doi":"10.37190/0208-4147.41.2.8","DOIUrl":"https://doi.org/10.37190/0208-4147.41.2.8","url":null,"abstract":"In this paper we prove that two heterogeneous parallel systems with independent exponentially distributed components are comparable via the star transform order while the comparison via the convex transform fails. The latter conclusion provides a partial answer to a problem that remained open for a decade.","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43122660","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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