. Let { X i ( t ) : t ∈ S ⊂ R d } i =1 , 2 ,...,n be independent copies of a stationary centered Gaussian field with almost surely smooth sample paths. In this paper, we are interested in the conjunction probability defined as P ( ∃ t ∈ S : X i ( t ) ≥ u, ∀ i = 1 , 2 , . . . , n ) for a given threshold level u . As u → ∞ , we will provide an asymptotic formula for the conjunction probability. This asymptotic formula is derived from the behaviour of the volume of the set of local maximum points. The proof relies on a result of Azaïs and Wschebor (2014) describing the shape of the excursion set of a stationary centered Gaussian field. Our result partially confirms the validity of the Euler characteristic method.
。设{X i (t): t∈S∧R d} i = 1,2，…，n是平稳中心高斯场的独立副本，样本路径几乎肯定是光滑的。在本文中，我们对定义为P(∃t∈S: X i (t)≥u，∀i = 1,2，…)的联结概率感兴趣。， n)表示给定阈值水平u。当u→∞时，我们将给出合取概率的渐近公式。这个渐近公式是由局部极大点集合的体积性质导出的。该证明依赖于Azaïs和Wschebor(2014)的结果，该结果描述了平稳中心高斯场的偏移集的形状。我们的结果部分地证实了欧拉特征方法的有效性。
{"title":"Asymptotic formula for the conjunction probability of smooth stationary Gaussian fields","authors":"Viet-Hung Pham","doi":"10.30757/alea.v20-29","DOIUrl":"https://doi.org/10.30757/alea.v20-29","url":null,"abstract":". Let { X i ( t ) : t ∈ S ⊂ R d } i =1 , 2 ,...,n be independent copies of a stationary centered Gaussian field with almost surely smooth sample paths. In this paper, we are interested in the conjunction probability defined as P ( ∃ t ∈ S : X i ( t ) ≥ u, ∀ i = 1 , 2 , . . . , n ) for a given threshold level u . As u → ∞ , we will provide an asymptotic formula for the conjunction probability. This asymptotic formula is derived from the behaviour of the volume of the set of local maximum points. The proof relies on a result of Azaïs and Wschebor (2014) describing the shape of the excursion set of a stationary centered Gaussian field. Our result partially confirms the validity of the Euler characteristic method.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69591775","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}
We study a $d$-dimensional random walk with exponentially distributed increments conditioned so that the components stay ordered (in the sense of Doob). We find explicitly a positive harmonic function $h$ for the killed process and then construct an ordered process using Doob's $h$-transform. Since these random walks are not nearest-neighbour, the harmonic function is not the Vandermonde determinant. The ordered process is related to the departure process of M/M/1 queues in tandem. We find asymptotics for the tail probabilities of the time until the components in exponential random walks become disordered and a local limit theorem. We find the distribution of the processes of smallest and largest particles as Fredholm determinants.
{"title":"Ordered exponential random walks","authors":"Denis Denisov, Will FitzGerald","doi":"10.30757/alea.v20-45","DOIUrl":"https://doi.org/10.30757/alea.v20-45","url":null,"abstract":"We study a $d$-dimensional random walk with exponentially distributed increments conditioned so that the components stay ordered (in the sense of Doob). We find explicitly a positive harmonic function $h$ for the killed process and then construct an ordered process using Doob's $h$-transform. Since these random walks are not nearest-neighbour, the harmonic function is not the Vandermonde determinant. The ordered process is related to the departure process of M/M/1 queues in tandem. We find asymptotics for the tail probabilities of the time until the components in exponential random walks become disordered and a local limit theorem. We find the distribution of the processes of smallest and largest particles as Fredholm determinants.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":"2016 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135843074","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}
. This paper is concerned with the asymptotic analysis of sojourn times of random fields with continuous sample paths. Under a very general framework we show that there is an interesting relationship between tail asymptotics of sojourn times and that of supremum. Moreover, we establish the uniform double-sum method to derive the tail asymptotics of sojourn times. In the literature, based on the pioneering research of S. Berman the sojourn times have been utilised to derive the tail asymptotics of supremum of Gaussian processes. In this paper we show that the opposite direction is even more fruitful, namely knowing the asymptotics of supremum of random processes and fields (in particular Gaussian) it is possible to establish the asymptotics of their sojourn times. We illustrate our findings considering i) two dimensional Gaussian random fields, ii) chi-process generated by stationary Gaussian processes and iii) stationary Gaussian queueing processes.
{"title":"Sojourn times of Gaussian and related random fields","authors":"K. Dȩbicki, E. Hashorva, Peng Liu, Z. Michna","doi":"10.30757/alea.v20-10","DOIUrl":"https://doi.org/10.30757/alea.v20-10","url":null,"abstract":". This paper is concerned with the asymptotic analysis of sojourn times of random fields with continuous sample paths. Under a very general framework we show that there is an interesting relationship between tail asymptotics of sojourn times and that of supremum. Moreover, we establish the uniform double-sum method to derive the tail asymptotics of sojourn times. In the literature, based on the pioneering research of S. Berman the sojourn times have been utilised to derive the tail asymptotics of supremum of Gaussian processes. In this paper we show that the opposite direction is even more fruitful, namely knowing the asymptotics of supremum of random processes and fields (in particular Gaussian) it is possible to establish the asymptotics of their sojourn times. We illustrate our findings considering i) two dimensional Gaussian random fields, ii) chi-process generated by stationary Gaussian processes and iii) stationary Gaussian queueing processes.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69591630","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}
A random walk on a group is noise sensitive if resampling every step independantly with a small probability results in an almost independant output. We precisely define two notions: $ell^1$-noise sensitivity and entropy noise sensitivity. Groups with one of these properties are necessarily Liouville. Homomorphisms to free abelian groups provide an obstruction to $ell^1$-noise sensitivity. We also provide examples of $ell^1$ and entropy noise sensitive random walks.
如果以小概率独立地对每一步重新采样会产生几乎独立的输出，则对组上的随机漫步是噪声敏感的。我们精确地定义了两个概念:$ well ^1$-噪声灵敏度和熵噪声灵敏度。具有这些性质之一的群必然是刘维尔群。自由阿贝尔群的同态对$ell^1$-噪声灵敏度有阻碍作用。我们还提供了$ well ^1$和熵噪声敏感随机漫步的例子。
{"title":"Noise sensitivity of random walks on groups","authors":"Itaï Benjamini, Jérémie Brieussel","doi":"10.30757/alea.v20-42","DOIUrl":"https://doi.org/10.30757/alea.v20-42","url":null,"abstract":"A random walk on a group is noise sensitive if resampling every step independantly with a small probability results in an almost independant output. We precisely define two notions: $ell^1$-noise sensitivity and entropy noise sensitivity. Groups with one of these properties are necessarily Liouville. Homomorphisms to free abelian groups provide an obstruction to $ell^1$-noise sensitivity. We also provide examples of $ell^1$ and entropy noise sensitive random walks. ","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135181363","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}
For moving average processes with random coefficients and heavy-tailed innovations that are weakly dependent in the sense of strong mixing and local dependence condition $D'$ we study joint functional convergence of partial sums and maxima. Under the assumption that all partial sums of the series of coefficients are a.s. bounded between zero and the sum of the series we derive a functional limit theorem in the space of $mathbb{R}^{2}$-valued c`{a}dl`{a}g functions on $[0, 1]$ with the Skorokhod weak $M_{2}$ topology.
{"title":"Joint functional convergence of partial sums and maxima for moving averages with weakly dependent heavy-tailed innovations and random coefficients","authors":"Danijel Krizmanić","doi":"10.30757/alea.v20-46","DOIUrl":"https://doi.org/10.30757/alea.v20-46","url":null,"abstract":"For moving average processes with random coefficients and heavy-tailed innovations that are weakly dependent in the sense of strong mixing and local dependence condition $D'$ we study joint functional convergence of partial sums and maxima. Under the assumption that all partial sums of the series of coefficients are a.s. bounded between zero and the sum of the series we derive a functional limit theorem in the space of $mathbb{R}^{2}$-valued c`{a}dl`{a}g functions on $[0, 1]$ with the Skorokhod weak $M_{2}$ topology.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135843449","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}
. Alternative proofs of the characterizations of the wide-sense geometric and of the Marshall-Olkin exponential distributions via monotone set functions are provided. In contrast to the ones presented in Shenkman (2017), which rely on the generative constructions of Arnold (1975) or Marshall and Olkin (1967) to establish that certain functions equipped with monotone parameters are proper survival functions, we aim herein to check that these candidates satisfy a set of well known necessary and sufficient analytical conditions. The major difficulty in such an approach consists in verifying that they do not infringe any of the so-called rectangle inequalities. Fortunately, a factorization shows that compliance is guaranteed as long as a finite number of very specific “basis” rectangle inequalities are not violated: a condition which is, by the very definition of the monotone parameters, trivially met.
。通过单调集合函数给出了广义几何分布和Marshall-Olkin指数分布表征的替代证明。Shenkman(2017)依靠Arnold(1975)或Marshall and Olkin(1967)的生成构造来确定某些配备单调参数的函数是适当的生存函数，与之相反，我们在这里的目的是检查这些候选函数是否满足一组众所周知的充分必要分析条件。这种方法的主要困难在于核实它们不违反任何所谓的矩形不等式。幸运的是，因式分解表明，只要不违反有限数量的非常特定的“基”矩形不等式，就可以保证遵从性:根据单调参数的定义，这个条件通常是满足的。
{"title":"Characterizations of multivariate distributions with limited memory revisited: An analytical approach","authors":"Natalia Shenkman","doi":"10.30757/alea.v20-39","DOIUrl":"https://doi.org/10.30757/alea.v20-39","url":null,"abstract":". Alternative proofs of the characterizations of the wide-sense geometric and of the Marshall-Olkin exponential distributions via monotone set functions are provided. In contrast to the ones presented in Shenkman (2017), which rely on the generative constructions of Arnold (1975) or Marshall and Olkin (1967) to establish that certain functions equipped with monotone parameters are proper survival functions, we aim herein to check that these candidates satisfy a set of well known necessary and sufficient analytical conditions. The major difficulty in such an approach consists in verifying that they do not infringe any of the so-called rectangle inequalities. Fortunately, a factorization shows that compliance is guaranteed as long as a finite number of very specific “basis” rectangle inequalities are not violated: a condition which is, by the very definition of the monotone parameters, trivially met.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69591375","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}
Adrián González Casanova, Lizbeth Peñaloza, Arno Siri-Jégousse
In this article, we introduce a random (directed) graph model for the simultaneous forwards and backwards description of a rather broad class of Cannings models with a seed bank mechanism. This provides a simple tool to establish a sampling duality in the finite population size, and obtain a path-wise embedding of the forward frequency process and the backward ancestral process. Further, it allows the derivation of limit theorems that generalize celebrated results by M"ohle to models with seed banks, and where it can be seen how the effect of seed banks affects the genealogies. The explicit graphical construction is a new tool to understand the subtle interplay of seed banks, reproduction and genetic drift in population genetics.
{"title":"Seed bank Cannings graphs: How dormancy smoothes random genetic drift","authors":"Adrián González Casanova, Lizbeth Peñaloza, Arno Siri-Jégousse","doi":"10.30757/alea.v20-43","DOIUrl":"https://doi.org/10.30757/alea.v20-43","url":null,"abstract":"In this article, we introduce a random (directed) graph model for the simultaneous forwards and backwards description of a rather broad class of Cannings models with a seed bank mechanism. This provides a simple tool to establish a sampling duality in the finite population size, and obtain a path-wise embedding of the forward frequency process and the backward ancestral process. Further, it allows the derivation of limit theorems that generalize celebrated results by M\"ohle to models with seed banks, and where it can be seen how the effect of seed banks affects the genealogies. The explicit graphical construction is a new tool to understand the subtle interplay of seed banks, reproduction and genetic drift in population genetics.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135843453","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}
. The Dirichlet-Ferguson measure is a cornerstone in nonparametric Bayesian statistics and the study of the distributional properties of expectations with respect to such measure is an important line of research initiated in Cifarelli and Regazzini (1979a,b) and still very active, see Letac and Piccioni (2018) and Lijoi and Prünster (2009). In this paper we provide explicit upper bounds for the d 3 , the d 2 and the convex distances between random vectors whose components are means of the Dirichlet-Ferguson measure and a random vector distributed according to the multivariate Gaussian law. These results are applied to the Gaussian approximation of linear transformations of random vectors with the Dirichlet distribution, yielding presumably optimal rates on the d 3 and the d 2 distances and presumably suboptimal rates on the convex and the Kolmogorov distances.
{"title":"Quantitative Multidimensional Central Limit Theorems for Means of the Dirichlet-Ferguson Measure","authors":"G. Torrisi","doi":"10.30757/alea.v20-30","DOIUrl":"https://doi.org/10.30757/alea.v20-30","url":null,"abstract":". The Dirichlet-Ferguson measure is a cornerstone in nonparametric Bayesian statistics and the study of the distributional properties of expectations with respect to such measure is an important line of research initiated in Cifarelli and Regazzini (1979a,b) and still very active, see Letac and Piccioni (2018) and Lijoi and Prünster (2009). In this paper we provide explicit upper bounds for the d 3 , the d 2 and the convex distances between random vectors whose components are means of the Dirichlet-Ferguson measure and a random vector distributed according to the multivariate Gaussian law. These results are applied to the Gaussian approximation of linear transformations of random vectors with the Dirichlet distribution, yielding presumably optimal rates on the d 3 and the d 2 distances and presumably suboptimal rates on the convex and the Kolmogorov distances.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69591329","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}
. As a well-known and important extension of the common Poisson model with an additional parameter, Conway-Maxwell-Poisson (CMP) distributions allow for describing under-and overdispersion in discrete data. Constituting a two-parameter exponential family, CMP distributions possess useful structural and statistical properties. However, the exponential family is not steep and maximum likelihood estimation may fail even for non-trivial data sets, which is different from the Poisson case, where maximum likelihood estimation only fails if all data outcomes are zero. Conditions are examined for existence and non-existence of maximum likelihood estimates in the full family as well as in subfamilies of CMP distributions, and several figures illustrate the problem.
{"title":"On the existence of maximum likelihood estimates for the parameters of the Conway-Maxwell-Poisson distribution","authors":"S. Bedbur, U. Kamps, A. Imm","doi":"10.30757/alea.v20-20","DOIUrl":"https://doi.org/10.30757/alea.v20-20","url":null,"abstract":". As a well-known and important extension of the common Poisson model with an additional parameter, Conway-Maxwell-Poisson (CMP) distributions allow for describing under-and overdispersion in discrete data. Constituting a two-parameter exponential family, CMP distributions possess useful structural and statistical properties. However, the exponential family is not steep and maximum likelihood estimation may fail even for non-trivial data sets, which is different from the Poisson case, where maximum likelihood estimation only fails if all data outcomes are zero. Conditions are examined for existence and non-existence of maximum likelihood estimates in the full family as well as in subfamilies of CMP distributions, and several figures illustrate the problem.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69591754","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}
We consider multidimensional random walks in pyramids, which by definition are cones formed by finite intersections of half-spaces. The main object of interest is the survival probability $mathbb{P}(tau>n)$, $tau$ denoting the first exit time from a fixed pyramid. When the drift belongs to the interior of the cone, the survival probability sequence converges to the non-exit probability $mathbb{P}(tau=infty)$, which is positive. In this note, we quantify the speed of convergence, and prove that the exponential rate of convergence may be computed by means of a certain min-max of the Laplace transform of the random walk increments. We illustrate our results with various examples.
{"title":"Random walks with drift inside a pyramid: convergence rate for the survival probability","authors":"Rodolphe Garbit, K. Raschel","doi":"10.30757/alea.v20-35","DOIUrl":"https://doi.org/10.30757/alea.v20-35","url":null,"abstract":"We consider multidimensional random walks in pyramids, which by definition are cones formed by finite intersections of half-spaces. The main object of interest is the survival probability $mathbb{P}(tau>n)$, $tau$ denoting the first exit time from a fixed pyramid. When the drift belongs to the interior of the cone, the survival probability sequence converges to the non-exit probability $mathbb{P}(tau=infty)$, which is positive. In this note, we quantify the speed of convergence, and prove that the exponential rate of convergence may be computed by means of a certain min-max of the Laplace transform of the random walk increments. We illustrate our results with various examples.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44216661","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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