This paper is to study the properties of eigenvalues and eigenvectors of high dimensional sample correlation matrices. We firstly improve the result of Jiang (2004); Xiao and Zhou (2010) and the Theorem 1 of Karoui (2009), both concerning the limiting spectral distribution and the extreme eigenvalues of sample correlation matrices, by allowing a more general fourth moment condition. Then, we establish a central limit theorem (CLT) for the linear statistics of the eigenvectors of large sample correlation matrices. We discover that the difference between the functional CLT of the sample covariance matrix and that of the sample correlation matrix is fundamentally influenced by the direction of a nonrandom projection vector. In the special case where the square root of the correlation matrix is identity, the difference will be determined by the sum of the fourth powers of the entries of the projection vector. These results also indicate that the eigenmatrix of sample correlation matrix is not asymptotic Haar if the underlying distribution is Gaussian. In other words, the normalization based on the sample variances affects the asymptotic properties of the eigenmatrix of the Wishart matrix. Furthermore, we establish a theorem concerning CLT for the linear statistics of the eigenvectors of large sample covariance matrices. This theorem improves the main result in Bai, Miao, and Pan (2007), which requires the assumption that the fourth moment of the underlying variable matches the one of Gaussian distribution, as well as Theorem 1.3 in Pan and Zhou (2008), which relaxes the Gaussian like fourth moment requirement but assumes the maximum entries of the projection vectors converge to 0 (i.e. the `∞ norms of the projection vectors converge to 0). We illustrate the usefulness of the theoretical results through an application in communications.
{"title":"Properties of eigenvalues and eigenvectors of large-dimensional sample correlation matrices","authors":"Yanqing Yin, Yanyuan Ma","doi":"10.1214/22-aap1802","DOIUrl":"https://doi.org/10.1214/22-aap1802","url":null,"abstract":"This paper is to study the properties of eigenvalues and eigenvectors of high dimensional sample correlation matrices. We firstly improve the result of Jiang (2004); Xiao and Zhou (2010) and the Theorem 1 of Karoui (2009), both concerning the limiting spectral distribution and the extreme eigenvalues of sample correlation matrices, by allowing a more general fourth moment condition. Then, we establish a central limit theorem (CLT) for the linear statistics of the eigenvectors of large sample correlation matrices. We discover that the difference between the functional CLT of the sample covariance matrix and that of the sample correlation matrix is fundamentally influenced by the direction of a nonrandom projection vector. In the special case where the square root of the correlation matrix is identity, the difference will be determined by the sum of the fourth powers of the entries of the projection vector. These results also indicate that the eigenmatrix of sample correlation matrix is not asymptotic Haar if the underlying distribution is Gaussian. In other words, the normalization based on the sample variances affects the asymptotic properties of the eigenmatrix of the Wishart matrix. Furthermore, we establish a theorem concerning CLT for the linear statistics of the eigenvectors of large sample covariance matrices. This theorem improves the main result in Bai, Miao, and Pan (2007), which requires the assumption that the fourth moment of the underlying variable matches the one of Gaussian distribution, as well as Theorem 1.3 in Pan and Zhou (2008), which relaxes the Gaussian like fourth moment requirement but assumes the maximum entries of the projection vectors converge to 0 (i.e. the `∞ norms of the projection vectors converge to 0). We illustrate the usefulness of the theoretical results through an application in communications.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49091572","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}
Seppäläinen and Valkó showed in [20] that for a suitable choice of parameters, the variance growth of the free energy of the stationary O’Connell-Yor polymer is governed by the exponent 2/3, characteristic of models in the KPZ universality class. We develop exact formulas based on Gaussian integration by parts to relate the cumulants of the free energy, logZθ n,t, to expectations of products of quenched cumulants of the time of the first jump from the boundary into the system, s0. We then use these formulas to obtain estimates for the k-th central moment of logZθ n,t as well as the k-th annealed moment of s0 for k > 2, with nearly optimal exponents (1/3)k + and (2/3)k + , respectively. As an application, we derive new high probability bounds for the distance between the polymer path and a straight line connecting the origin to the endpoint of the path.
{"title":"Central moments of the free energy of the stationary O’Connell–Yor polymer","authors":"C. Noack, Philippe Sosoe","doi":"10.1214/21-aap1744","DOIUrl":"https://doi.org/10.1214/21-aap1744","url":null,"abstract":"Seppäläinen and Valkó showed in [20] that for a suitable choice of parameters, the variance growth of the free energy of the stationary O’Connell-Yor polymer is governed by the exponent 2/3, characteristic of models in the KPZ universality class. We develop exact formulas based on Gaussian integration by parts to relate the cumulants of the free energy, logZθ n,t, to expectations of products of quenched cumulants of the time of the first jump from the boundary into the system, s0. We then use these formulas to obtain estimates for the k-th central moment of logZθ n,t as well as the k-th annealed moment of s0 for k > 2, with nearly optimal exponents (1/3)k + and (2/3)k + , respectively. As an application, we derive new high probability bounds for the distance between the polymer path and a straight line connecting the origin to the endpoint of the path.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48849556","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}
We study the sample covariance matrix for real-valued data with general population covariance, as well as MANOVA-type covariance estimators in variance components models under null hypotheses of global sphericity. In the limit as matrix dimensions increase proportionally, the asymptotic spectra of such estimators may have multiple disjoint intervals of support, possibly intersecting the negative half line. We show that the distribution of the extremal eigenvalue at each regular edge of the support has a GOE Tracy-Widom limit. Our proof extends a comparison argument of Ji Oon Lee and Kevin Schnelli, replacing a continuous Green function flow by a discrete Lindeberg swapping scheme.
{"title":"TRACY-WIDOM AT EACH EDGE OF REAL COVARIANCE AND MANOVA ESTIMATORS.","authors":"Zhou Fan, Iain M Johnstone","doi":"","DOIUrl":"","url":null,"abstract":"<p><p>We study the sample covariance matrix for real-valued data with general population covariance, as well as MANOVA-type covariance estimators in variance components models under null hypotheses of global sphericity. In the limit as matrix dimensions increase proportionally, the asymptotic spectra of such estimators may have multiple disjoint intervals of support, possibly intersecting the negative half line. We show that the distribution of the extremal eigenvalue at each regular edge of the support has a GOE Tracy-Widom limit. Our proof extends a comparison argument of Ji Oon Lee and Kevin Schnelli, replacing a continuous Green function flow by a discrete Lindeberg swapping scheme.</p>","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":"32 4","pages":"2967-3003"},"PeriodicalIF":1.8,"publicationDate":"2022-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9410589/pdf/nihms-1829912.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"33443472","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Lemma 3.1 asserts that Eyn[Z(̃yn)] − Exn[Z(̃xn)] = (n−1/2) and Eyn[Z(̃yn)] > Exn[Z(̃xn)] for all sufficiently large n. Our proof was not correct: As Benjamin Gunby and Xiaoyu He pointed out to us, we missed four terms in the computation of equation (3.3). Those terms contribute a negative amount, so the proof is more delicate. Here is a correct proof. The intuition behind the result is that a string with a defect of the type we consider, namely, a 10 in a string of 01’s, is likely to cause more 11’s in the trace than a string without the defect. Since the defect in yn is shifted to the right as compared to the defect in xn, the defect of yn is slightly more likely to “fall into” the test window { 2np + 1 , . . . , 2np + √npq } of the trace than is the defect of xn. More precisely, the difference in probability is of order n−1/2. In the proof below, we make this intuition rigorous. PROOF. We assume throughout the proof that k ∈ { 2np + 1 , . . . , 2np + √npq }. Let E(m,k) denote the event that bit m in the input string is copied to position k in the trace. First observe that
{"title":"Erratum to “Lower bounds for trace reconstruction”","authors":"N. Holden, R. Lyons","doi":"10.1214/22-aap1827","DOIUrl":"https://doi.org/10.1214/22-aap1827","url":null,"abstract":"Lemma 3.1 asserts that Eyn[Z(̃yn)] − Exn[Z(̃xn)] = (n−1/2) and Eyn[Z(̃yn)] > Exn[Z(̃xn)] for all sufficiently large n. Our proof was not correct: As Benjamin Gunby and Xiaoyu He pointed out to us, we missed four terms in the computation of equation (3.3). Those terms contribute a negative amount, so the proof is more delicate. Here is a correct proof. The intuition behind the result is that a string with a defect of the type we consider, namely, a 10 in a string of 01’s, is likely to cause more 11’s in the trace than a string without the defect. Since the defect in yn is shifted to the right as compared to the defect in xn, the defect of yn is slightly more likely to “fall into” the test window { 2np + 1 , . . . , 2np + √npq } of the trace than is the defect of xn. More precisely, the difference in probability is of order n−1/2. In the proof below, we make this intuition rigorous. PROOF. We assume throughout the proof that k ∈ { 2np + 1 , . . . , 2np + √npq }. Let E(m,k) denote the event that bit m in the input string is copied to position k in the trace. First observe that","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46337326","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}
This work concerns the asymptotic behavior of solutions to a critical fluid model for a data communication network, where file sizes are generally distributed and the network operates under a fair bandwidth sharing policy, chosen from the family of (weighted) α-fair policies introduced by Mo and Walrand [18]. Solutions of the fluid model are measure-valued functions of time. Under law of large numbers scaling, Gromoll and Williams [8] proved that these solutions approximate dynamic solutions of a flow level model for congestion control in data communication networks, introduced by Massoulié and Roberts [17]. In a recent work [6], we proved stability of the strictly subcritical version of this fluid model under mild assumptions. In the current work, we study the asymptotic behavior (as time goes to infinity) of solutions of the critical fluid model, in which the nominal load on each network resource is less than or equal to its capacity and at least one resource is fully loaded. For this we introduce a new Lyapunov function, inspired by the work of Kelly and Williams [14], Mulvany et al. [19] and Paganini et al. [20]. Using this, under moderate conditions on the file size distributions, we prove that critical fluid model solutions converge uniformly to the set of invariant states as time goes to infinity, when started in suitable relatively compact sets. We expect that this result will play a key role in developing a diffusion approximation for the critically loaded flow level model of Massoulié and Roberts [17]. Furthermore, the techniques developed here may be useful for studying other stochastic network models with resource sharing. ∗Research supported in part by NSF grants DMS-1206772 and DMS-1712974, and the Charles Lee Powell Foundation. A preliminary form of some of the material in this paper was featured in the Le Cam Lecture delivered by RJW at the IMS Annual Meeting held in Vilnius, Lithuania, in July 2018. †Department of Mathematics, University of California, San Diego, 9500 Gilman Drive, La Jolla CA 92093-0112. Email: yif051@ucsd.edu. ‡Department of Mathematics, University of California, San Diego, 9500 Gilman Drive, La Jolla CA 92093-0112. Email: rjwilliams@ucsd.edu. MSC2020 Mathematics Subject Classification: Primary 60F99, 60K30, 90B10; Secondary 60J25, 60K25, 90B18.
{"title":"Asymptotic behavior of a critical fluid model for bandwidth sharing with general file size distributions","authors":"Yingjia Fu, Ruth J. Williams","doi":"10.1214/21-aap1723","DOIUrl":"https://doi.org/10.1214/21-aap1723","url":null,"abstract":"This work concerns the asymptotic behavior of solutions to a critical fluid model for a data communication network, where file sizes are generally distributed and the network operates under a fair bandwidth sharing policy, chosen from the family of (weighted) α-fair policies introduced by Mo and Walrand [18]. Solutions of the fluid model are measure-valued functions of time. Under law of large numbers scaling, Gromoll and Williams [8] proved that these solutions approximate dynamic solutions of a flow level model for congestion control in data communication networks, introduced by Massoulié and Roberts [17]. In a recent work [6], we proved stability of the strictly subcritical version of this fluid model under mild assumptions. In the current work, we study the asymptotic behavior (as time goes to infinity) of solutions of the critical fluid model, in which the nominal load on each network resource is less than or equal to its capacity and at least one resource is fully loaded. For this we introduce a new Lyapunov function, inspired by the work of Kelly and Williams [14], Mulvany et al. [19] and Paganini et al. [20]. Using this, under moderate conditions on the file size distributions, we prove that critical fluid model solutions converge uniformly to the set of invariant states as time goes to infinity, when started in suitable relatively compact sets. We expect that this result will play a key role in developing a diffusion approximation for the critically loaded flow level model of Massoulié and Roberts [17]. Furthermore, the techniques developed here may be useful for studying other stochastic network models with resource sharing. ∗Research supported in part by NSF grants DMS-1206772 and DMS-1712974, and the Charles Lee Powell Foundation. A preliminary form of some of the material in this paper was featured in the Le Cam Lecture delivered by RJW at the IMS Annual Meeting held in Vilnius, Lithuania, in July 2018. †Department of Mathematics, University of California, San Diego, 9500 Gilman Drive, La Jolla CA 92093-0112. Email: yif051@ucsd.edu. ‡Department of Mathematics, University of California, San Diego, 9500 Gilman Drive, La Jolla CA 92093-0112. Email: rjwilliams@ucsd.edu. MSC2020 Mathematics Subject Classification: Primary 60F99, 60K30, 90B10; Secondary 60J25, 60K25, 90B18.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45772590","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 consider a mean field particle systems whose confinement potentials have many local minima. We establish some explicit and sharp estimates of the spectral gap and logarithmic Sobolev constants uniform in the number of particles. The uniform Poincaré inequality is based on the work of Ledoux [20] and the uniform logarithmic Sobolev inequality is based on Zegarlinski’s theorem for Gibbs measures, both combined with an explicit estimate of the Lipschitz norm of the Poisson operator for a single particle from [29]. The logarithmic Sobolev inequality then implies the exponential convergence in entropy of the McKean-Vlasov equation with an explicit rate, We need here weaker conditions than the results of [10] (by means of the displacement convexity approach), [21, 22] (by Bakry-Emery’s technique) or the recent work [9] (by dissipation of the Wasserstein distance).
{"title":"Uniform Poincaré and logarithmic Sobolev inequalities for mean field particle systems","authors":"A. Guillin, Wei Liu, Liming Wu, Chao Zhang","doi":"10.1214/21-aap1707","DOIUrl":"https://doi.org/10.1214/21-aap1707","url":null,"abstract":"In this paper we consider a mean field particle systems whose confinement potentials have many local minima. We establish some explicit and sharp estimates of the spectral gap and logarithmic Sobolev constants uniform in the number of particles. The uniform Poincaré inequality is based on the work of Ledoux [20] and the uniform logarithmic Sobolev inequality is based on Zegarlinski’s theorem for Gibbs measures, both combined with an explicit estimate of the Lipschitz norm of the Poisson operator for a single particle from [29]. The logarithmic Sobolev inequality then implies the exponential convergence in entropy of the McKean-Vlasov equation with an explicit rate, We need here weaker conditions than the results of [10] (by means of the displacement convexity approach), [21, 22] (by Bakry-Emery’s technique) or the recent work [9] (by dissipation of the Wasserstein distance).","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48265138","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 give a natural extension of the antithetic multilevel Monte Carlo (MLMC) estimator for a multi-dimensional diffusion introduced by Giles and Szpruch [15] by considering the permutation between m Brownian increments, m ≥ 2, instead of using two increments as in the original paper. Our aim is to study the asymptotic behavior of the weak errors involved in this new algorithm. Among the obtained results, we prove that the error between on the one hand the average of the Milstein scheme without Lévy area and its σ-antithetic version build on the finer grid and on the other hand the coarse approximation stably converges in distribution with a rate of order 1. We also prove that the error between the Milstein scheme without Lévy area and its σ-antithetic version stably converges in distribution with a rate of order 1/2. More precisely, we have a functional limit theorem on the asymptotic behavior of the joined distribution of these errors based on a triangular array approach (see e.g. Jacod [20]). Thanks to this result, we establish a central limit theorem of Lindeberg-Feller type for the antithetic MLMC estimator. The time complexity of the algorithm is analyzed.
{"title":"Central limit theorem for the antithetic multilevel Monte Carlo method","authors":"M. Ben Alaya, Ahmed Kebaier, T. Ngo","doi":"10.1214/21-aap1726","DOIUrl":"https://doi.org/10.1214/21-aap1726","url":null,"abstract":"In this paper, we give a natural extension of the antithetic multilevel Monte Carlo (MLMC) estimator for a multi-dimensional diffusion introduced by Giles and Szpruch [15] by considering the permutation between m Brownian increments, m ≥ 2, instead of using two increments as in the original paper. Our aim is to study the asymptotic behavior of the weak errors involved in this new algorithm. Among the obtained results, we prove that the error between on the one hand the average of the Milstein scheme without Lévy area and its σ-antithetic version build on the finer grid and on the other hand the coarse approximation stably converges in distribution with a rate of order 1. We also prove that the error between the Milstein scheme without Lévy area and its σ-antithetic version stably converges in distribution with a rate of order 1/2. More precisely, we have a functional limit theorem on the asymptotic behavior of the joined distribution of these errors based on a triangular array approach (see e.g. Jacod [20]). Thanks to this result, we establish a central limit theorem of Lindeberg-Feller type for the antithetic MLMC estimator. The time complexity of the algorithm is analyzed.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48443365","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}
Antonio Blanca, P. Caputo, D. Parisi, A. Sinclair, Eric Vigoda
{"title":"Entropy decay in the Swendsen–Wang dynamics on Zd","authors":"Antonio Blanca, P. Caputo, D. Parisi, A. Sinclair, Eric Vigoda","doi":"10.1214/21-aap1702","DOIUrl":"https://doi.org/10.1214/21-aap1702","url":null,"abstract":"","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45671817","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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