Pub Date : 2022-01-01DOI: 10.37190/0208-4147.00060
Nobuaki Naganuma
. Azmoodeh et al. established a criterion regarding convergence of the second and other even moments of random variables in a Wiener chaos with fixed order guaranteeing the central convergence of the random variables. This was a major step in studies of the fourth moment theorem. In this paper, we provide further generalizations of the fourth moment theorem by building on their ideas. More precisely, further criteria implying central convergence are provided: (i) the convergence of the fourth and any other even moment, (ii) the convergence of the sixth and some other even moments.
{"title":"Generalizations of the fourth moment theorem","authors":"Nobuaki Naganuma","doi":"10.37190/0208-4147.00060","DOIUrl":"https://doi.org/10.37190/0208-4147.00060","url":null,"abstract":". Azmoodeh et al. established a criterion regarding convergence of the second and other even moments of random variables in a Wiener chaos with fixed order guaranteeing the central convergence of the random variables. This was a major step in studies of the fourth moment theorem. In this paper, we provide further generalizations of the fourth moment theorem by building on their ideas. More precisely, further criteria implying central convergence are provided: (i) the convergence of the fourth and any other even moment, (ii) the convergence of the sixth and some other even moments.","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69996351","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 : 2022-01-01DOI: 10.37190/0208-4147.00032
Aaron Abrams, E. Babson, H. Landau, Zeph Landau, James Pommersheim
. We give an elementary proof of a result due to Diaconis and Saloff-Coste (1994) that families of symmetric simple random walks on Cayley graphs of abelian groups with a bound on the number of generators never have sharp cutoff. Here convergence to the stationary distribution is measured in the total variation norm. This is a situation of bounded degree and no expansion; sharp cutoff (or the cutoff phenomenon) has been shown to occur in families such as random walks on a hypercube (Diaco-nis, 1996) in which the degree is unbounded as well as on a random regular graph where the degree is fixed, but there is expansion (Diaconis and Saloff-Coste, 1993).
{"title":"No cutoff for circulants: an elementary proof","authors":"Aaron Abrams, E. Babson, H. Landau, Zeph Landau, James Pommersheim","doi":"10.37190/0208-4147.00032","DOIUrl":"https://doi.org/10.37190/0208-4147.00032","url":null,"abstract":". We give an elementary proof of a result due to Diaconis and Saloff-Coste (1994) that families of symmetric simple random walks on Cayley graphs of abelian groups with a bound on the number of generators never have sharp cutoff. Here convergence to the stationary distribution is measured in the total variation norm. This is a situation of bounded degree and no expansion; sharp cutoff (or the cutoff phenomenon) has been shown to occur in families such as random walks on a hypercube (Diaco-nis, 1996) in which the degree is unbounded as well as on a random regular graph where the degree is fixed, but there is expansion (Diaconis and Saloff-Coste, 1993).","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69996743","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 : 2022-01-01DOI: 10.37190/0208-4147.00018
Xin Liao, Zuoxiang Peng, S. Nadarajah
. Liao, Peng and Nadarajah [ J. Appl. Probab. 50 (2013), 900–907] derived asymptotic expansions for the partial maximum of a random sam-ple from the logarithmic skew-normal distribution. Here, we derive asymptotic expansions for moments of the partial maximum using optimal norm-ing constants. These expansions can be used to deduce convergence rates of moments of the normalized maxima to the moments of the correspond-ing extreme value distribution. A numerical study is made to compare the actual values of moments with their asymptotics, which shows that the convergence is exceedingly slow, and adjustment is needed whenever we use the limits to replace moments of the partial maximum.
{"title":"Expansions for moments of logarithmic skew-normal extremes","authors":"Xin Liao, Zuoxiang Peng, S. Nadarajah","doi":"10.37190/0208-4147.00018","DOIUrl":"https://doi.org/10.37190/0208-4147.00018","url":null,"abstract":". Liao, Peng and Nadarajah [ J. Appl. Probab. 50 (2013), 900–907] derived asymptotic expansions for the partial maximum of a random sam-ple from the logarithmic skew-normal distribution. Here, we derive asymptotic expansions for moments of the partial maximum using optimal norm-ing constants. These expansions can be used to deduce convergence rates of moments of the normalized maxima to the moments of the correspond-ing extreme value distribution. A numerical study is made to compare the actual values of moments with their asymptotics, which shows that the convergence is exceedingly slow, and adjustment is needed whenever we use the limits to replace moments of the partial maximum.","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69996692","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 : 2022-01-01DOI: 10.37190/0208-4147.00088
J. Rosínski, J. Szulga
{"title":"In Memoriam: Wojbor Andrzej Woyczyński (1943-2021)","authors":"J. Rosínski, J. Szulga","doi":"10.37190/0208-4147.00088","DOIUrl":"https://doi.org/10.37190/0208-4147.00088","url":null,"abstract":"","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69996764","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 : 2022-01-01DOI: 10.37190/0208-4147.00071
P. Matuła
. Let ( X n ) n ∈ N and ( Y n ) n ∈ N be two sequences of i.i.d
． 设(X n) n∈n， (Y n) n∈n为两个序列i.i.d
{"title":"A remark on the exact laws of large numbers for ratios of independent random variables","authors":"P. Matuła","doi":"10.37190/0208-4147.00071","DOIUrl":"https://doi.org/10.37190/0208-4147.00071","url":null,"abstract":". Let ( X n ) n ∈ N and ( Y n ) n ∈ N be two sequences of i.i.d","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69996419","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 : 2022-01-01DOI: 10.37190/0208-4147.00036
T. Byczkowski, T. Żak, J. Małecki
. We prove an analogue of Ehrhard’s inequality for the two-dimensional isotropic Cauchy measure. In contrast to the Gaussian case, the inequality is not valid for non-convex sets. We provide the proof for rectangles which are symmetric with respect to one coordinate axis.
{"title":"Ehrhard-type inequality for isotropic Cauchy distribution on the plane","authors":"T. Byczkowski, T. Żak, J. Małecki","doi":"10.37190/0208-4147.00036","DOIUrl":"https://doi.org/10.37190/0208-4147.00036","url":null,"abstract":". We prove an analogue of Ehrhard’s inequality for the two-dimensional isotropic Cauchy measure. In contrast to the Gaussian case, the inequality is not valid for non-convex sets. We provide the proof for rectangles which are symmetric with respect to one coordinate axis.","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69996284","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 : 2022-01-01DOI: 10.37190/0208-4147.00050
C. Pelekis, R. Fokkink, S. Papavassiliou
. Let S and X be independent random variables, assuming values in the set of non-negative integers, and suppose further that both E ( S ) and E ( X ) are integers satisfying E ( S ) › E ( X ) . We establish a sufficient condition for the tail probability P ( S › E ( S )) to be larger than the tail P ( S + X › E ( S + X )) , when the mean of S is equal to the mode.
． 设S和X为独立随机变量，设值在非负整数集合中，并进一步假设E (S)和E (X)都是满足E (S)›E (X)的整数。当S的均值等于模态时，我们建立了尾部概率P (S›E (S))大于尾部概率P (S + X›E (S + X))的充分条件。
{"title":"On the monotonicity of tail probabilities","authors":"C. Pelekis, R. Fokkink, S. Papavassiliou","doi":"10.37190/0208-4147.00050","DOIUrl":"https://doi.org/10.37190/0208-4147.00050","url":null,"abstract":". Let S and X be independent random variables, assuming values in the set of non-negative integers, and suppose further that both E ( S ) and E ( X ) are integers satisfying E ( S ) › E ( X ) . We establish a sufficient condition for the tail probability P ( S › E ( S )) to be larger than the tail P ( S + X › E ( S + X )) , when the mean of S is equal to the mode.","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69996296","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 : 2022-01-01DOI: 10.37190/0208-4147.00010
Z. Jurek
{"title":"Exponential bounds of ruin probabilities for non-homogeneous risk models","authors":"Z. Jurek","doi":"10.37190/0208-4147.00010","DOIUrl":"https://doi.org/10.37190/0208-4147.00010","url":null,"abstract":"","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69996682","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 : 2021-10-03DOI: 10.37190/0208-4147.00043
Nikita Karagodin
We prove a limit theorem on the convergence of the distributions of the scaled last exit time over a slowly moving nonlinear boundary for a class of Gaussian stationary processes. The limit is a double exponential (Gumbel) distribution.
{"title":"A limit theorem for the last exit time over a moving nonlinear boundary for a Gaussian process","authors":"Nikita Karagodin","doi":"10.37190/0208-4147.00043","DOIUrl":"https://doi.org/10.37190/0208-4147.00043","url":null,"abstract":"We prove a limit theorem on the convergence of the distributions of the scaled last exit time over a slowly moving nonlinear boundary for a class of Gaussian stationary processes. The limit is a double exponential (Gumbel) distribution.","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":"38 9","pages":""},"PeriodicalIF":0.3,"publicationDate":"2021-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41270676","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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