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1. Local limit theorems for densities in Orlicz spaces

   Bobkov, S. G. (October 2019, Journal of mathematical sciences)

   Necessary and sufficient conditions for the validity of the central limit theorem for densities are considered with respect to the norms in Orlicz spaces. The obtained characterization unites several results due to Gnedenko and Kolmogorov (uniform local limit theorem), Prokhorov (convergence in total variation) and Barron (entropic central limit theorem). 

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2. Strictly subgaussian probability distributions

   https://doi.org/10.1214/24-EJP1120  

   Bobkov, S G; Chistyakov, G P; Götze, F (January 2024, Electronic Journal of Probability)

   We explore probability distributions on the real line whose Laplace transform admits an upper bound of subgaussian type known as strict subgaussianity. One class in this family corresponds to entire characteristic functions having only real zeros in the complex plane. Using Hadamard’s factorization theorem, we extend this class and propose new sufficient conditions for strict subgaussianity in terms of location of zeros of the associated characteristic functions. The second part of this note deals with Laplace transforms of strictly subgaussian distributions with periodic components. This class contains interesting examples, for which the central limit theorem with respect to the Rényi entropy divergence of infinite order holds. 

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3. Esscher transform and the central limit theorem

   https://doi.org/10.1016/j.jfa.2025.110999  

   Bobkov, Sergey G; Götze, Friedrich (September 2025, Journal of Functional Analysis)

   The paper is devoted to the investigation of Esscher’s transform on high dimensional Euclidean spaces in the light of its application to the central limit theorem. With this tool, we explore necessary and sufficient conditions of normal approximation for normalized sums of i.i.d. random vectors in terms of the Rényi divergence of infinite order, extending recent one dimensional results. 

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4. A PDE hierarchy for directed polymers in random environments

   https://doi.org/10.1088/1361-6544/ac23b7  

   Gu, Yu; Henderson, Christopher (September 2021, Nonlinearity)

   Abstract For a Brownian directed polymer in a Gaussian random environment, with q ( t , ⋅) denoting the quenched endpoint density and Q n ( t , x 1 , … , x n ) = E [ q ( t , x 1 ) … q ( t , x n ) ] , we derive a hierarchical PDE system satisfied by { Q n } n ⩾ 1 . We present two applications of the system: (i) we compute the generator of { μ t ( d x ) = q ( t , x ) d x } t ⩾ 0 for some special functionals, where { μ t ( d x ) } t ⩾ 0 is viewed as a Markov process taking values in the space of probability measures; (ii) in the high temperature regime with d ⩾ 3, we prove a quantitative central limit theorem for the annealed endpoint distribution of the diffusively rescaled polymer path. We also study a nonlocal diffusion-reaction equation motivated by the generator and establish a super-diffusive O ( t 2/3 ) scaling. 

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5. 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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