{"title":"Refined behaviour of a conditioned random walk in the large deviations regime","authors":"Søren Asmussen, Peter W. Glynn","doi":"10.3150/23-bej1601","DOIUrl":"https://doi.org/10.3150/23-bej1601","url":null,"abstract":"","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139684555","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we deal with a Markov chain on a measurable state space ( X , X ) which has a transition kernel P admitting an aperiodic small-set S and satisfying the standard geometric-drift condition. Under these assumptions, there exists α 0 ∈ (0 , 1] such that PV α 0 ≤ δ α 0 V α 0 + ν ( V α 0 )1 S . Hence P is V α 0 − geometrically ergodic and its “second eigenvalue” ϱ α 0 provides the best rate of convergence. Setting R := P − ν ( · )1 S and Γ := { λ ∈ C , δ α 0 < | λ | < 1 } , ϱ α 0 is shown to satisfy, either ϱ α 0 = max (cid:8) | λ | : λ ∈ Γ , (cid:80) + ∞ k =1 λ − k ν ( R k − 1 1 S ) = 1 (cid:9) if this set is not empty, or ϱ α 0 ≤ δ α 0 . Actually the set is finite in the first case and is composed by the spectral values of P in Γ. The second case occurs when P has no spectral value in Γ. Moreover, a bound of the operator-norm of ( zI − P ) − 1 allows us to derive an explicit formula for the multiplicative constant in the rate of convergence, which can be evaluated provided that any information of the “second eigenvalue” is available. Such numerical computation is carried out for a classical family of reflected random walks. Moreover we obtain a simple and explicit bound of the operator-norm of ( I − P + π ( · )1 X ) − 1 involved in the definition of the so-called fundamental solution to Poisson’s equation. This allows us to specify the location of the eigenvalues of P and, then, to obtain a general bound on ϱ α 0 . The reversible case is also discussed. In particular, the bound of ϱ α 0 obtained for positive reversible Markov kernels is the expected one, and numerical illustrations are proposed for the Metropolis-Hastings algorithm and for the Gaussian autoregressive Markov chain. The bound for the operator-norm of ( I − P + π ( · )1 X ) − 1 is derived from an estimate, only depending on δ α 0 , of the operator-norm of ( I − R ) − 1 which provides another way to get a solution to Poisson’s equation. This estimate is also shown to be of greatest interest to generalize the error bounds obtained for perturbed discrete and atomic Markov chains in [LL18] to the case of general geometrically ergodic Markov chains. These error estimates are the simplest that can be expected in this context. All the estimates in this work are expressed in the standard V α 0 − weighted operator norm.
本文处理的是可测状态空间 ( X , X ) 上的马尔可夫链，它有一个过渡核 P，允许一个非周期性小集 S，并满足标准几何漂移条件。在这些假设条件下，存在 α 0∈ (0 , 1]，使得 PV α 0 ≤ δ α 0 V α 0 + ν ( V α 0 )1 S。因此，P 是 V α 0 - 几何遍历，其 "第二特征值" ϱ α 0 提供了最佳收敛速率。设 R := P - ν ( - )1 S 和 Γ := { λ ∈ C , δ α 0 < | λ | < 1 } , ϱ α 0 可提供最佳收敛率。α 0 = max (cid:8) | λ | : λ ∈ Γ , (cid:80) + ∞ k =1 λ - k ν ( R k - 1 1 S ) = 1 (cid:9) 如果这个集合不空，或者ϱ α 0 ≤ δ α 0。实际上，在第一种情况下，集合是有限的，由 P 在 Γ 中的谱值组成。第二种情况是 P 在 Γ 中没有谱值。此外，( zI - P ) - 1 的算子规范约束允许我们推导出收敛速度乘法常数的明确公式，只要有 "第二特征值 "的任何信息，就可以对其进行评估。这种数值计算是针对经典的反射随机游走族进行的。此外，我们还得到了 ( I - P + π ( - ) 1 X ) 的算子正的一个简单明了的约束。- 1 的算子矩阵，它涉及所谓泊松方程基本解的定义。这样，我们就能明确 P 的特征值位置，进而得到 ϱ α 0 的一般约束。我们还讨论了可逆情况。特别是，对正可逆马尔科夫核得到的ϱ α 0 的约束是预期的约束，并提出了 Metropolis-Hastings 算法和高斯自回归马尔科夫链的数值说明。( I - P + π ( - )1 X ) 的算子规范的约束- 1 的算子正态的估计值推导而得，该估计值仅取决于 ( I - R ) - 1 的算子正态，它提供了另一种求解泊松方程的方法。这个估计值对于将 [LL18] 中针对扰动离散马尔可夫链和原子马尔可夫链得到的误差边界推广到一般几何遍历马尔可夫链的情况也是最有意义的。在这种情况下，这些误差估计是最简单的。这项工作中的所有估计都用标准 V α 0 - 加权算子规范表示。
{"title":"Explicit bounds for spectral theory of geometrically ergodic Markov kernels and applications","authors":"L. Hervé, James Ledoux","doi":"10.3150/23-bej1609","DOIUrl":"https://doi.org/10.3150/23-bej1609","url":null,"abstract":"In this paper, we deal with a Markov chain on a measurable state space ( X , X ) which has a transition kernel P admitting an aperiodic small-set S and satisfying the standard geometric-drift condition. Under these assumptions, there exists α 0 ∈ (0 , 1] such that PV α 0 ≤ δ α 0 V α 0 + ν ( V α 0 )1 S . Hence P is V α 0 − geometrically ergodic and its “second eigenvalue” ϱ α 0 provides the best rate of convergence. Setting R := P − ν ( · )1 S and Γ := { λ ∈ C , δ α 0 < | λ | < 1 } , ϱ α 0 is shown to satisfy, either ϱ α 0 = max (cid:8) | λ | : λ ∈ Γ , (cid:80) + ∞ k =1 λ − k ν ( R k − 1 1 S ) = 1 (cid:9) if this set is not empty, or ϱ α 0 ≤ δ α 0 . Actually the set is finite in the first case and is composed by the spectral values of P in Γ. The second case occurs when P has no spectral value in Γ. Moreover, a bound of the operator-norm of ( zI − P ) − 1 allows us to derive an explicit formula for the multiplicative constant in the rate of convergence, which can be evaluated provided that any information of the “second eigenvalue” is available. Such numerical computation is carried out for a classical family of reflected random walks. Moreover we obtain a simple and explicit bound of the operator-norm of ( I − P + π ( · )1 X ) − 1 involved in the definition of the so-called fundamental solution to Poisson’s equation. This allows us to specify the location of the eigenvalues of P and, then, to obtain a general bound on ϱ α 0 . The reversible case is also discussed. In particular, the bound of ϱ α 0 obtained for positive reversible Markov kernels is the expected one, and numerical illustrations are proposed for the Metropolis-Hastings algorithm and for the Gaussian autoregressive Markov chain. The bound for the operator-norm of ( I − P + π ( · )1 X ) − 1 is derived from an estimate, only depending on δ α 0 , of the operator-norm of ( I − R ) − 1 which provides another way to get a solution to Poisson’s equation. This estimate is also shown to be of greatest interest to generalize the error bounds obtained for perturbed discrete and atomic Markov chains in [LL18] to the case of general geometrically ergodic Markov chains. These error estimates are the simplest that can be expected in this context. All the estimates in this work are expressed in the standard V α 0 − weighted operator norm.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139686173","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A note on the empty balls of a critical super-Brownian motion","authors":"Shuxiong Zhang, Jie Xiong","doi":"10.3150/22-bej1564","DOIUrl":"https://doi.org/10.3150/22-bej1564","url":null,"abstract":"","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139687211","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Concentration of measure bounds for matrix-variate data with missing values","authors":"Shuheng Zhou","doi":"10.3150/23-bej1594","DOIUrl":"https://doi.org/10.3150/23-bej1594","url":null,"abstract":"","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139688104","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On bivariate distributions of the local time of Itô-McKean diffusions","authors":"Jacek Jakubowski, Maciej Wisniewolski","doi":"10.3150/23-bej1595","DOIUrl":"https://doi.org/10.3150/23-bej1595","url":null,"abstract":"","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139685544","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Entrywise limit theorems for eigenvectors of signal-plus-noise matrix models with weak signals","authors":"Fangzheng Xie","doi":"10.3150/23-bej1602","DOIUrl":"https://doi.org/10.3150/23-bej1602","url":null,"abstract":"","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139687608","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
M. Casanellas, Marina Garrote-López, Piotr Zwiernik
MARTA CASANELLAS1,2, MARINA GARROTE-LÓPEZ3 and PIOTR ZWIERNIK4 1Institut de Matemàtiques de la UPC-BarcelonaTech (IMTech). Universitat Politècnica de Catalunya, Barcelona, Spain, E-mail: marta.casanellas@upc.edu 2Centre de Recerca Matemàtica, Barcelona, Spain 3Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany, E-mail: marina.garrote@upc.edu 4Department of Statistical Sciences, University of Toronto, Canada, E-mail: piotr.zwiernik@utoronto.ca
MARTA CASANELLAS1,2, MARINA GARROTE-LÓPEZ3 and PIOTR ZWIERNIK4 1Institut de Matemàtiques de la UPC-BarcelonaTech (IMTech).Universitat Politècnica de Catalunya, Barcelona, Spain, E-mail: marta.casanellas@upc.edu 2Centre de Recerca Matemàtica, Barcelona, Spain 3Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany, E-mail: marina.garrote@upc.edu 4Department of Statistical Sciences, University of Toronto, Canada, E-mail: piotr.zwiernik@utoronto.ca
{"title":"Identifiability in robust estimation of tree structured models","authors":"M. Casanellas, Marina Garrote-López, Piotr Zwiernik","doi":"10.3150/22-bej1477","DOIUrl":"https://doi.org/10.3150/22-bej1477","url":null,"abstract":"MARTA CASANELLAS1,2, MARINA GARROTE-LÓPEZ3 and PIOTR ZWIERNIK4 1Institut de Matemàtiques de la UPC-BarcelonaTech (IMTech). Universitat Politècnica de Catalunya, Barcelona, Spain, E-mail: marta.casanellas@upc.edu 2Centre de Recerca Matemàtica, Barcelona, Spain 3Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany, E-mail: marina.garrote@upc.edu 4Department of Statistical Sciences, University of Toronto, Canada, E-mail: piotr.zwiernik@utoronto.ca","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139687725","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Linear and nonlinear signal detection and estimation in high-dimensional nonparametric regression under weak sparsity","authors":"Kin Yap Cheung, Stephen M. S. Lee, Xiaoya Xu","doi":"10.3150/23-bej1611","DOIUrl":"https://doi.org/10.3150/23-bej1611","url":null,"abstract":"","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139685797","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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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