Pub Date : 2018-07-30DOI: 10.19195/0208-4147.38.1.4
Chunming Zhao
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{"title":"Extremes of order statistics of stationary Gaussian processes","authors":"Chunming Zhao","doi":"10.19195/0208-4147.38.1.4","DOIUrl":"https://doi.org/10.19195/0208-4147.38.1.4","url":null,"abstract":"Tu wpisz tekst","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2018-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46051840","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 : 2018-07-30DOI: 10.19195/0208-4147.38.1.8
T. Rogala, Ł. Stettner
In the paper we solve a system of Bellman equations for finite horizon continuous time terminal utility maximization problem with general càdlàg bid and ask prices. We assume that we have a restricted number of transactions at time moments we choose. The main result of the paper says that we can find a regular version of solutions to the system of Bellman equations, which enables us to find the form of nearly optimal strategies.
{"title":"Bellman equations for terminal utility maximization with general bid and ask prices","authors":"T. Rogala, Ł. Stettner","doi":"10.19195/0208-4147.38.1.8","DOIUrl":"https://doi.org/10.19195/0208-4147.38.1.8","url":null,"abstract":"In the paper we solve a system of Bellman equations for finite horizon continuous time terminal utility maximization problem with general càdlàg bid and ask prices. We assume that we have a restricted number of transactions at time moments we choose. The main result of the paper says that we can find a regular version of solutions to the system of Bellman equations, which enables us to find the form of nearly optimal strategies.","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2018-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41704667","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 : 2018-07-30DOI: 10.19195/0208-4147.38.1.3
P. Kozyra, T. Rychlik
We describe a method of calculating sharp lower and upper bounds on the expectations of linear combinations of kth records expressed in the Gini mean difference units of the original i.i.d. observations. In particular, we provide sharp lower and upper bounds on the expectations of kth records and their differences. We also present the families of distributions which attain the bounds in the limit.
{"title":"Sharp bounds on the expectations of linear combinations of kth records expressed in the Gini mean difference units","authors":"P. Kozyra, T. Rychlik","doi":"10.19195/0208-4147.38.1.3","DOIUrl":"https://doi.org/10.19195/0208-4147.38.1.3","url":null,"abstract":"We describe a method of calculating sharp lower and upper bounds on the expectations of linear combinations of kth records expressed in the Gini mean difference units of the original i.i.d. observations. In particular, we provide sharp lower and upper bounds on the expectations of kth records and their differences. We also present the families of distributions which attain the bounds in the limit.","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2018-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44181348","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 : 2018-07-30DOI: 10.19195/0208-4147.38.1.1
F. Proïa
We tackle the stationarity issue of an autoregressive path with a polynomial trend, and generalize some aspects of the LMC test, the testing procedure of Leybourne and McCabe. First, we show that it is possible to get the asymptotic distribution of the test statistic under the null hypothesis of trend-stationarity as well as under the alternative of nonstationarity for any polynomial trend of order r. Then, we explain the reason why the LMC test, and by extension the KPSS test, does not reject the null hypothesis of trend-stationarity, mistakenly, when the random walk is generated by a unit root located at −1.We also observe it on simulated data and correct the procedure. Finally, we describe some useful stochastic processes that appear in our limiting distributions.
{"title":"Stationarity against integration in the autoregressive process with polynomial trend","authors":"F. Proïa","doi":"10.19195/0208-4147.38.1.1","DOIUrl":"https://doi.org/10.19195/0208-4147.38.1.1","url":null,"abstract":"We tackle the stationarity issue of an autoregressive path with a polynomial trend, and generalize some aspects of the LMC test, the testing procedure of Leybourne and McCabe. First, we show that it is possible to get the asymptotic distribution of the test statistic under the null hypothesis of trend-stationarity as well as under the alternative of nonstationarity for any polynomial trend of order r. Then, we explain the reason why the LMC test, and by extension the KPSS test, does not reject the null hypothesis of trend-stationarity, mistakenly, when the random walk is generated by a unit root located at −1.We also observe it on simulated data and correct the procedure. Finally, we describe some useful stochastic processes that appear in our limiting distributions.","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2018-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45774118","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 : 2018-07-30DOI: 10.19195/0208-4147.38.1.13
W. Pestman, F. Tuerlinckx, W. Vanpaemel
In this paper the seminal Jeffreys–Lindley paradox is regarded from a mathematical point of view. We show that in certain scenarios the paradox may emerge in a reverse direction.
{"title":"A reverse to the Jeffreys–Lindley paradox","authors":"W. Pestman, F. Tuerlinckx, W. Vanpaemel","doi":"10.19195/0208-4147.38.1.13","DOIUrl":"https://doi.org/10.19195/0208-4147.38.1.13","url":null,"abstract":"In this paper the seminal Jeffreys–Lindley paradox is regarded from a mathematical point of view. We show that in certain scenarios the paradox may emerge in a reverse direction.","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2018-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42934716","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 : 2018-07-30DOI: 10.19195/0208-4147.38.1.12
M. Azimmohseni, A. Soltani, M. Khalafi, Naeemeh Akbari Ghalesary
In this article, we introduce a weighted periodogram in the class of smoothed periodograms as a consistent estimator for the spectral density matrix of a periodically correlated process. We derive its limiting distribution that appears to be a certain finite linear combination of Wishart distribution. We also provide numerical derivations for our smoothed periodogram and exhibit its asymptotic consistency using simulated data.
{"title":"A consistent estimator for spectral density matrix of a discrete time periodically correlated process","authors":"M. Azimmohseni, A. Soltani, M. Khalafi, Naeemeh Akbari Ghalesary","doi":"10.19195/0208-4147.38.1.12","DOIUrl":"https://doi.org/10.19195/0208-4147.38.1.12","url":null,"abstract":"In this article, we introduce a weighted periodogram in the class of smoothed periodograms as a consistent estimator for the spectral density matrix of a periodically correlated process. We derive its limiting distribution that appears to be a certain finite linear combination of Wishart distribution. We also provide numerical derivations for our smoothed periodogram and exhibit its asymptotic consistency using simulated data.","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2018-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41615531","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 : 2018-06-11DOI: 10.37190/0208-4147.41.1.1
Tord Sjödin
Let $Y$ be a standard Gamma(k) distributed random variable, $k>0$, and let $X$ be an independent positive random variable. We prove that if $X$ has a hyperbolically monotone density of order $k$ ($HM_k$), then the distributions of $Ycdot X$ and $Y/X$ are generalized gamma convolutions (GGC). This result extends results of Roynette et al. and Behme and Bondesson, who treated respectively the cases $k=1$ and $k$ an integer. We give a proof that covers all $k>0$ and gives explicit formulas for the relevant functions that extend those found by Behme and Bondesson in the integer case.
设$Y$为标准的Gamma(k)分布随机变量，$k>0$，设$X$为独立的正随机变量。证明了如果$X$具有$k$ ($HM_k$)阶的双曲单调密度，则$Ycdot X$和$Y/X$的分布是广义伽马卷积(GGC)。这个结果扩展了Roynette et al.和Behme and Bondesson的结果，他们分别处理了$k=1$和$k$为整数的情况。我们给出了一个涵盖所有$k> $的证明，并给出了扩展Behme和Bondesson在整数情况下发现的相关函数的显式公式。
{"title":"On Mixtures of Gamma distributions, distributions with hyperbolically monotone densities and Generalized Gamma Convolutions (GGC)","authors":"Tord Sjödin","doi":"10.37190/0208-4147.41.1.1","DOIUrl":"https://doi.org/10.37190/0208-4147.41.1.1","url":null,"abstract":"Let $Y$ be a standard Gamma(k) distributed random variable, $k>0$, and let $X$ be an independent positive random variable. We prove that if $X$ has a hyperbolically monotone density of order $k$ ($HM_k$), then the distributions of $Ycdot X$ and $Y/X$ are generalized gamma convolutions (GGC). This result extends results of Roynette et al. and Behme and Bondesson, who treated respectively the cases $k=1$ and $k$ an integer. We give a proof that covers all $k>0$ and gives explicit formulas for the relevant functions that extend those found by Behme and Bondesson in the integer case.","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2018-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44013372","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 : 2018-04-13DOI: 10.19195/0208-4147.39.2.10
M. Olszewski
We give sharp two-sided estimates for the functions gMt, x, y and gMt, x, y − gt, x, y, where gMt, x, y are the transition probability densities of the reflected Brownian motion on an Mcomplex of order M ∈ Z of an unbounded planar simple nested fractal and gt, x, y are the transition probability densities of the “free” Brownian motion on this fractal. This is done for a large class of planar simple nested fractals with the good labeling property.
{"title":"Estimates of the transition densities for the reflected Brownian motion on simple nested fractals","authors":"M. Olszewski","doi":"10.19195/0208-4147.39.2.10","DOIUrl":"https://doi.org/10.19195/0208-4147.39.2.10","url":null,"abstract":"We give sharp two-sided estimates for the functions gMt, x, y and gMt, x, y − gt, x, y, where gMt, x, y are the transition probability densities of the reflected Brownian motion on an Mcomplex of order M ∈ Z of an unbounded planar simple nested fractal and gt, x, y are the transition probability densities of the “free” Brownian motion on this fractal. This is done for a large class of planar simple nested fractals with the good labeling property.","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2018-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42689555","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 : 2018-02-02DOI: 10.37190/0208-4147.41.2.7
W. Bednorz, R. Lochowski, Rafał Martynek
We estimate up to universal constants tails of symmetric and totally asymmetric $alpha$-stable distributions in terms of functions of the parameters of these distributions.
根据对称和完全不对称$alpha$稳定分布的参数函数，我们估计了这些分布的普遍常数尾。
{"title":"On tails of symmetric and totally asymmetric alpha-stable distributions","authors":"W. Bednorz, R. Lochowski, Rafał Martynek","doi":"10.37190/0208-4147.41.2.7","DOIUrl":"https://doi.org/10.37190/0208-4147.41.2.7","url":null,"abstract":"We estimate up to universal constants tails of symmetric and totally asymmetric $alpha$-stable distributions in terms of functions of the parameters of these distributions.","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2018-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46848842","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 : 2017-09-06DOI: 10.37190/0208-4147.40.1.10
P. Kevei
We prove a Tauberian theorem for the Laplace--Stieltjes transform and Karamata-type theorems in the framework of regularly log-periodic functions. As an application we determine the exact tail behavior of fixed points of certain type smoothing transforms.
{"title":"Regularly log-periodic functions and some applications","authors":"P. Kevei","doi":"10.37190/0208-4147.40.1.10","DOIUrl":"https://doi.org/10.37190/0208-4147.40.1.10","url":null,"abstract":"We prove a Tauberian theorem for the Laplace--Stieltjes transform and Karamata-type theorems in the framework of regularly log-periodic functions. As an application we determine the exact tail behavior of fixed points of certain type smoothing transforms.","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2017-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48967531","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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