Victor Chernozhukov, Denis Chetverikov, Yuta Koike
In this paper, we derive new, nearly optimal bounds for the Gaussian approximation to scaled averages of n independent high-dimensional centered random vectors X1,…,Xn over the class of rectangles in the case when the covariance matrix of the scaled average is nondegenerate. In the case of bounded Xi’s, the implied bound for the Kolmogorov distance between the distribution of the scaled average and the Gaussian vector takes the form C(Bn2log3d/n)1/2logn, where d is the dimension of the vectors and Bn is a uniform envelope constant on components of Xi’s. This bound is sharp in terms of d and Bn, and is nearly (up to logn) sharp in terms of the sample size n. In addition, we show that similar bounds hold for the multiplier and empirical bootstrap approximations. Moreover, we establish bounds that allow for unbounded Xi’s, formulated solely in terms of moments of Xi’s. Finally, we demonstrate that the bounds can be further improved in some special smooth and moment-constrained cases.
{"title":"Nearly optimal central limit theorem and bootstrap approximations in high dimensions","authors":"Victor Chernozhukov, Denis Chetverikov, Yuta Koike","doi":"10.1214/22-aap1870","DOIUrl":"https://doi.org/10.1214/22-aap1870","url":null,"abstract":"In this paper, we derive new, nearly optimal bounds for the Gaussian approximation to scaled averages of n independent high-dimensional centered random vectors X1,…,Xn over the class of rectangles in the case when the covariance matrix of the scaled average is nondegenerate. In the case of bounded Xi’s, the implied bound for the Kolmogorov distance between the distribution of the scaled average and the Gaussian vector takes the form C(Bn2log3d/n)1/2logn, where d is the dimension of the vectors and Bn is a uniform envelope constant on components of Xi’s. This bound is sharp in terms of d and Bn, and is nearly (up to logn) sharp in terms of the sample size n. In addition, we show that similar bounds hold for the multiplier and empirical bootstrap approximations. Moreover, we establish bounds that allow for unbounded Xi’s, formulated solely in terms of moments of Xi’s. Finally, we demonstrate that the bounds can be further improved in some special smooth and moment-constrained cases.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135104464","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}
Rodrigo A. Bazaes, Chiranjib Mukherjee, A. Ramírez, S. Saglietti
In 2003, Varadhan ( Comm. Pure Appl. Math. 56 (2003) 1222–1245) developed a robust method for proving quenched and averaged large deviations for random walks in a uniformly elliptic and i.i.d. environment (RWRE) on Z d . One fundamental question which remained open was to determine when the quenched and averaged large deviation rate functions agree, and when they do not. In this article we show that for RWRE in uniformly elliptic and i.i.d. environment in d ≥ 4, the two rate functions agree on any compact set contained in the interior of their domain which does not contain the origin, provided that the disorder of the environment is sufficiently low. Our result provides a new formulation which encompasses a set of sufficient conditions under which these rate functions agree without assuming that the RWRE is ballistic (see ( Probab. Theory Related Fields 149 (2011) 463–491)), satisfies a CLT or even a law of large numbers ( Electron. Commun. Probab. 7 (2002)191–197; Ann. Probab. 36 (2008) 728–738). Also, the equality of rate functions is not restricted to neighborhoods around given points, as long as the disorder of the environment is kept low. One of the novelties of our approach is the introduction of an auxiliary random walk in a deterministic environment which is itself ballistic (regardless of the actual RWRE behavior) and whose large deviation properties approximate those of the original RWRE in a robust manner, even if the original RWRE is not ballistic itself.
2003年，Varadhan（Comm.Pure Appl.Math.56（2003）1222–1245）开发了一种稳健的方法，用于证明Z d上均匀椭圆和i.i.d.环境（RWRE）中随机游动的淬火和平均大偏差。一个悬而未决的基本问题是确定淬火和平均大偏差率函数何时一致，何时不一致。在本文中，我们证明了对于一致椭圆和d≥4的i.i.d.环境中的RWRE，只要环境的无序度足够低，两个速率函数在其域内部包含的任何不包含原点的紧集上都是一致的。我们的结果提供了一个新的公式，它包含了一组充分的条件，在这些条件下，这些速率函数一致，而不假设RWRE是弹道的（见（Probab.Theory Related Fields 149（2011）463–491）），满足CLT甚至大数律（Electron.Commun.Probab.7（2002）191–197；Ann.Probab。36（2008）728-738）。此外，速率函数的相等性不限于给定点周围的邻域，只要环境的无序性保持在较低水平即可。我们方法的新颖之处之一是在确定性环境中引入了辅助随机行走，该环境本身就是弹道的（与实际RWRE行为无关），并且其大偏差特性以稳健的方式近似于原始RWRE的大偏差特性，即使原始RWRE本身不是弹道的。
{"title":"Quenched and averaged large deviations for random walks in random environments: The impact of disorder","authors":"Rodrigo A. Bazaes, Chiranjib Mukherjee, A. Ramírez, S. Saglietti","doi":"10.1214/22-aap1864","DOIUrl":"https://doi.org/10.1214/22-aap1864","url":null,"abstract":"In 2003, Varadhan ( Comm. Pure Appl. Math. 56 (2003) 1222–1245) developed a robust method for proving quenched and averaged large deviations for random walks in a uniformly elliptic and i.i.d. environment (RWRE) on Z d . One fundamental question which remained open was to determine when the quenched and averaged large deviation rate functions agree, and when they do not. In this article we show that for RWRE in uniformly elliptic and i.i.d. environment in d ≥ 4, the two rate functions agree on any compact set contained in the interior of their domain which does not contain the origin, provided that the disorder of the environment is sufficiently low. Our result provides a new formulation which encompasses a set of sufficient conditions under which these rate functions agree without assuming that the RWRE is ballistic (see ( Probab. Theory Related Fields 149 (2011) 463–491)), satisfies a CLT or even a law of large numbers ( Electron. Commun. Probab. 7 (2002)191–197; Ann. Probab. 36 (2008) 728–738). Also, the equality of rate functions is not restricted to neighborhoods around given points, as long as the disorder of the environment is kept low. One of the novelties of our approach is the introduction of an auxiliary random walk in a deterministic environment which is itself ballistic (regardless of the actual RWRE behavior) and whose large deviation properties approximate those of the original RWRE in a robust manner, even if the original RWRE is not ballistic itself.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42911546","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":"The spatial Λ-Fleming–Viot process in a random environment","authors":"A. Klimek, T. Rosati","doi":"10.1214/22-aap1871","DOIUrl":"https://doi.org/10.1214/22-aap1871","url":null,"abstract":"","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46895076","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}
One of the fundamental assumptions in stochastic control of continuous time processes is that the dynamics of the underlying (diffu-sion) process is known. This is, however, usually obviously not fulfilled in practice. On the other hand, over the last decades, a rich theory for nonparametric estimation of the drift (and volatility) for continuous time processes has been developed. The aim of this paper is bringing together techniques from stochastic control with methods from statistics for stochastic processes to find a way to both learn the dynamics of the underlying process and control in a reasonable way at the same time. More precisely, we study a long-term average impulse control problem, a stochastic version of the classical Faustmann timber harvesting problem. One of the problems that immediately arises is an exploration-exploitation dilemma as is well known for problems in machine learning. We propose a way to deal with this issue by combining exploration and exploitation periods in a suitable way. Our main finding is that this construction can be based on the rates of convergence of estimators for the invariant density. Using this, we obtain that the average cumulated regret is of uniform order O ( T − 1 / 3 ).
{"title":"Nonparametric learning for impulse control problems—Exploration vs. exploitation","authors":"S. Christensen, C. Strauch","doi":"10.1214/22-aap1849","DOIUrl":"https://doi.org/10.1214/22-aap1849","url":null,"abstract":"One of the fundamental assumptions in stochastic control of continuous time processes is that the dynamics of the underlying (diffu-sion) process is known. This is, however, usually obviously not fulfilled in practice. On the other hand, over the last decades, a rich theory for nonparametric estimation of the drift (and volatility) for continuous time processes has been developed. The aim of this paper is bringing together techniques from stochastic control with methods from statistics for stochastic processes to find a way to both learn the dynamics of the underlying process and control in a reasonable way at the same time. More precisely, we study a long-term average impulse control problem, a stochastic version of the classical Faustmann timber harvesting problem. One of the problems that immediately arises is an exploration-exploitation dilemma as is well known for problems in machine learning. We propose a way to deal with this issue by combining exploration and exploitation periods in a suitable way. Our main finding is that this construction can be based on the rates of convergence of estimators for the invariant density. Using this, we obtain that the average cumulated regret is of uniform order O ( T − 1 / 3 ).","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2023-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44358595","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}
(When the dimension d is clear from the context, we omit it from the notation, writing e.g. mt for mt(d), etc.) When d = 1, Bramson [5] proved the convergence in distribution of maxv∈NtX (v) t − mt(1), and the limit was identified by Lalley and Selke [10] to be the limit of a certain derivative martingale. It is not hard to deduce from their results and methods (see, e.g., [15, Thm. 1.1]) that, when d= 1,
{"title":"The maximum of branching Brownian motion in Rd","authors":"Yujin H. Kim, E. Lubetzky, O. Zeitouni","doi":"10.1214/22-aap1848","DOIUrl":"https://doi.org/10.1214/22-aap1848","url":null,"abstract":"(When the dimension d is clear from the context, we omit it from the notation, writing e.g. mt for mt(d), etc.) When d = 1, Bramson [5] proved the convergence in distribution of maxv∈NtX (v) t − mt(1), and the limit was identified by Lalley and Selke [10] to be the limit of a certain derivative martingale. It is not hard to deduce from their results and methods (see, e.g., [15, Thm. 1.1]) that, when d= 1,","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2023-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49028917","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}
Abstract. Consider a multi-type branching process in a random environment, whose reproduction law of generation n depends on the random environment at time n, unlike a constant distribution assumed in the Galton-Watson process. The famous Kesten-Stigum theorem for a supercritical multi-type Galton-Watson process gives a precise description of the exponential increasing rate of the population size via a criterion for the non-degeneracy of the fundamental martingale. Finding the corresponding result in the random environment case is a longstanding problem. For the single-type case the problem has been solved by Athreya and Karlin (1971) and Tanny (1988), but for the multi-type case it has been open for 50 years. Here we solve this problem in the typical case, by constructing a suitable martingale which reduces to the fundamental one in the constat environment case, and by establishing a criterion for the non-degeneracy of its limit. The convergence in law of the direction of the branching process is also considered. Our results open ways in establishing other limit theorems, such as law of large numbers, central limit theorems, Berry-Essen bound, and large deviation results.
{"title":"A Kesten–Stigum type theorem for a supercritical multitype branching process in a random environment","authors":"I. Grama, Quansheng Liu, Erwan Pin","doi":"10.1214/22-aap1840","DOIUrl":"https://doi.org/10.1214/22-aap1840","url":null,"abstract":"Abstract. Consider a multi-type branching process in a random environment, whose reproduction law of generation n depends on the random environment at time n, unlike a constant distribution assumed in the Galton-Watson process. The famous Kesten-Stigum theorem for a supercritical multi-type Galton-Watson process gives a precise description of the exponential increasing rate of the population size via a criterion for the non-degeneracy of the fundamental martingale. Finding the corresponding result in the random environment case is a longstanding problem. For the single-type case the problem has been solved by Athreya and Karlin (1971) and Tanny (1988), but for the multi-type case it has been open for 50 years. Here we solve this problem in the typical case, by constructing a suitable martingale which reduces to the fundamental one in the constat environment case, and by establishing a criterion for the non-degeneracy of its limit. The convergence in law of the direction of the branching process is also considered. Our results open ways in establishing other limit theorems, such as law of large numbers, central limit theorems, Berry-Essen bound, and large deviation results.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2023-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43167226","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 quantify the behaviour at large scales of the beta coalescent Π = {Π(t), t ≥ 0} with parameters a, b > 0. Specifically, we study the rescaled block size spectrum of Π(t) and of its restriction Πn(t) to {1, . . . , n}. Our main result is a Law of Large Numbers type of result if Π comes down from infinity. In the case of Kingman’s coalescent the derivation of this so-called hydrodynamic limit has been known since the work of Smoluchowski [30]. We extend Smoluchowski’s result to beta coalescents and show that if Π comes down from infinity both rescaled spectra
{"title":"Large-scale behaviour and hydrodynamic limit of beta coalescents","authors":"Luke Miller, Helmut H. Pitters","doi":"10.1214/22-aap1782","DOIUrl":"https://doi.org/10.1214/22-aap1782","url":null,"abstract":"We quantify the behaviour at large scales of the beta coalescent Π = {Π(t), t ≥ 0} with parameters a, b > 0. Specifically, we study the rescaled block size spectrum of Π(t) and of its restriction Πn(t) to {1, . . . , n}. Our main result is a Law of Large Numbers type of result if Π comes down from infinity. In the case of Kingman’s coalescent the derivation of this so-called hydrodynamic limit has been known since the work of Smoluchowski [30]. We extend Smoluchowski’s result to beta coalescents and show that if Π comes down from infinity both rescaled spectra","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48967718","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}
C. Landim, Carlos G. Pacheco, S. Sethuraman, Jianfei Xue
With the recent developments on nonlinear SPDE’s, where smoothing of rough noises is needed, one is naturally led to study interacting particle systems whose macroscopic evolution is described by these equations and which possess an in-built smoothing. In this article, our main results are to derive regularized versions of the ill-posed one dimensional SPDE ∂tρ = 1 2 ∆Φ(ρ)− 2∇ ( W ′Φ(ρ) ) , where the spatial white noise W ′ is replaced by a regularization W ′ ε, as quenched and annealed hydrodynamic limits of zero-range interacting particle systems in ε-regularized Sinai-type random environments. Some computations are also made about annealed mean hydrodynamic limits in unregularized Sinai-type random environments with respect to independent particles.
{"title":"On a nonlinear SPDE derived from a hydrodynamic limit in a Sinai-type random environment","authors":"C. Landim, Carlos G. Pacheco, S. Sethuraman, Jianfei Xue","doi":"10.1214/22-aap1813","DOIUrl":"https://doi.org/10.1214/22-aap1813","url":null,"abstract":"With the recent developments on nonlinear SPDE’s, where smoothing of rough noises is needed, one is naturally led to study interacting particle systems whose macroscopic evolution is described by these equations and which possess an in-built smoothing. In this article, our main results are to derive regularized versions of the ill-posed one dimensional SPDE ∂tρ = 1 2 ∆Φ(ρ)− 2∇ ( W ′Φ(ρ) ) , where the spatial white noise W ′ is replaced by a regularization W ′ ε, as quenched and annealed hydrodynamic limits of zero-range interacting particle systems in ε-regularized Sinai-type random environments. Some computations are also made about annealed mean hydrodynamic limits in unregularized Sinai-type random environments with respect to independent particles.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47342627","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 analyse the rate of convergence of a system of N interacting particles with mean-field rank-based interaction in the drift coefficient and constant diffusion coefficient. We first adapt arguments by Kolli and Shkolnikov [22] to check trajectorial propagation of chaos with optimal rate N−1/2 to the associated stochastic differential equations nonlinear in the sense of McKean. We next relax the assumptions needed by Bossy [6] to check the convergence in L (R) with rate O ( 1 √ N + h ) of the empirical cumulative distribution function of the Euler discretization with step h of the particle system to the solution of a one dimensional viscous scalar conservation law. Last, we prove that the bias of this stochastic particle method behaves as O ( 1 N + h ) . We provide numerical results which confirm our theoretical estimates.
本文分析了在漂移系数和恒定扩散系数下，具有平均场秩相互作用的N粒子相互作用系统的收敛速度。我们首先采用Kolli和Shkolnikov[22]的论点，对McKean意义上的非线性随机微分方程的最优速率N−1/2混沌的轨迹传播进行了检验。接下来，我们放宽了Bossy[6]检验粒子系统步长为h的欧拉离散的经验累积分布函数在L (R)以速率O(1√N + h)收敛到一维粘性标量守恒律解所需的假设。最后，我们证明了这种随机粒子方法的偏差表现为O (1 N + h)。我们提供的数值结果证实了我们的理论估计。
{"title":"Weak and strong error analysis for mean-field rank-based particle approximations of one-dimensional viscous scalar conservation laws","authors":"Oumaima Bencheikh, B. Jourdain","doi":"10.1214/21-aap1776","DOIUrl":"https://doi.org/10.1214/21-aap1776","url":null,"abstract":"In this paper, we analyse the rate of convergence of a system of N interacting particles with mean-field rank-based interaction in the drift coefficient and constant diffusion coefficient. We first adapt arguments by Kolli and Shkolnikov [22] to check trajectorial propagation of chaos with optimal rate N−1/2 to the associated stochastic differential equations nonlinear in the sense of McKean. We next relax the assumptions needed by Bossy [6] to check the convergence in L (R) with rate O ( 1 √ N + h ) of the empirical cumulative distribution function of the Euler discretization with step h of the particle system to the solution of a one dimensional viscous scalar conservation law. Last, we prove that the bias of this stochastic particle method behaves as O ( 1 N + h ) . We provide numerical results which confirm our theoretical estimates.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44524155","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 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":null,"pages":null},"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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：