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   https://doi.org/10.1007/s10959-023-01241-0  

   Peligrad, Magda (May 2023, Journal of Theoretical Probability)

   Fill, James Allen (Ed.)

   In this paper, we give sufficient conditions for the almost sure central limit theorem started at a point, known under the name of quenched central limit theorem. This is achieved by using a new idea of conditioning with respect to both the past and the future of the Markov chain. As applications, we provide a new sufficient projective condition for the quenched CLT. 

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2. Universality for the Conjugate Gradient and MINRES Algorithms on Sample Covariance Matrices

   https://doi.org/10.1002/cpa.22081  

   Paquette, Elliot; Trogdon, Thomas (May 2023, Communications on Pure and Applied Mathematics)

   Abstract We present a probabilistic analysis of two Krylov subspace methods for solving linear systems. We prove a central limit theorem for norms of the residual vectors that are produced by the conjugate gradient and MINRES algorithms when applied to a wide class of sample covariance matrices satisfying some standard moment conditions. The proof involves establishing a four‐moment theorem for the so‐called spectral measure, implying, in particular, universality for the matrix produced by the Lanczos iteration. The central limit theorem then implies an almost‐deterministic iteration count for the iterative methods in question. © 2022 Wiley Periodicals LLC. 

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3. Central limit theorem for the complex eigenvalues of Gaussian random matrices

   https://doi.org/10.1214/24-ECP585  

   Goel, Advay; Lopatto, Patrick; Xie, Xiaoyu (January 2024, Electronic Communications in Probability)

   We establish a central limit theorem for the eigenvalue counting function of a matrix of real Gaussian random variables. 

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4. Generalized kernel distance covariance in high dimensions: non-null CLTs and power universality

   https://doi.org/10.1093/imaiai/iaae017  

   Han, Qiyang; Shen, Yandi (July 2024, Information and Inference: A Journal of the IMA)

   Abstract Distance covariance is a popular dependence measure for two random vectors $$X$$ and $$Y$$ of possibly different dimensions and types. Recent years have witnessed concentrated efforts in the literature to understand the distributional properties of the sample distance covariance in a high-dimensional setting, with an exclusive emphasis on the null case that $$X$$ and $$Y$$ are independent. This paper derives the first non-null central limit theorem for the sample distance covariance, and the more general sample (Hilbert–Schmidt) kernel distance covariance in high dimensions, in the distributional class of $(X,Y)$ with a separable covariance structure. The new non-null central limit theorem yields an asymptotically exact first-order power formula for the widely used generalized kernel distance correlation test of independence between $$X$$ and $$Y$$. The power formula in particular unveils an interesting universality phenomenon: the power of the generalized kernel distance correlation test is completely determined by $$n\cdot \operatorname{dCor}^{2}(X,Y)/\sqrt{2}$$ in the high-dimensional limit, regardless of a wide range of choices of the kernels and bandwidth parameters. Furthermore, this separation rate is also shown to be optimal in a minimax sense. The key step in the proof of the non-null central limit theorem is a precise expansion of the mean and variance of the sample distance covariance in high dimensions, which shows, among other things, that the non-null Gaussian approximation of the sample distance covariance involves a rather subtle interplay between the dimension-to-sample ratio and the dependence between $$X$$ and $$Y$$. 

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5. Tempered functional time series

   https://doi.org/10.1111/jtsa.12667  

   Sabzikar, Farzad; Kokoszka, Piotr (November 2022, Journal of Time Series Analysis)

   We propose a broad class of models for time series of curves (functions) that can be used to quantify near long‐range dependence or near unit root behavior. We establish fundamental properties of these models and rates of consistency for the sample mean function and the sample covariance operator. The latter plays a role analogous to sample cross‐covariances for multivariate time series, but is far more important in the functional setting because its eigenfunctions are used in principal component analysis, which is a major tool in functional data analysis. It is used for dimension reduction of feature extraction. We also establish a central limit theorem for functions following our model. Both the consistency rates and the normalizations in the Central Limit Theorem (CLT) are nonstandard. They reflect the local unit root behavior and the long memory structure at moderate lags. 

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