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作者：Chhor, Julien; Sigalla, Suzanne; Tsybakov, Alexandre B.
作者单位：Universite de Toulouse; Universite Toulouse 1 Capitole; Toulouse School of Economics; Institut Polytechnique de Paris; Ecole Polytechnique; ENSAE Paris
摘要：We study benign overfitting in the setting of nonparametric regression under mean squared risk, and on the scale of H & ouml;lder classes. We construct a local polynomial estimator of the regression function that is minimax optimal on a H & ouml;lder class with any given smoothness, and that is a continuous function interpolating the set of observations with high probability. The key element of the construction is the use of singular kernels. Moreover, we prove that adaptation to unknown smoot...
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作者：Agresti, Antonio; Veraar, Mark
作者单位：Institute of Science & Technology - Austria; Delft University of Technology
摘要：In this paper we introduce the critical variational setting for parabolic stochastic evolution equations of quasi- or semi-linear type. Our results improve many of the abstract results in the classical variational setting. In particular, we are able to replace the usual weak or local monotonicity condition by a more flexible local Lipschitz condition. Moreover, the usual growth conditions on the multiplicative noise are weakened considerably. Our new setting provides general conditions under w...
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作者：Cipolloni, Giorgio; Erdos, Laszlo; Schroder, Dominik
作者单位：Princeton University; Institute of Science & Technology - Austria; Swiss Federal Institutes of Technology Domain; ETH Zurich
摘要：We prove that the mesoscopic linear statistics Sigma(i)f (n(a)(sigma(i) - z(0))) of the eigenvalues {sigma(i)}(i) of large nxn non-Hermitian random matrices with complex centred i.i.d. entries are asymptotically Gaussian for any H-0(2) -functions f around any point z0 in the bulk of the spectrum on any mesoscopic scale 0 < a < 1/2. This extends our previous result (Cipolloni et al. in Commun Pure Appl Math, 2019. arXiv:1912.04100), that was valid on the macroscopic scale, a = 0, to cover the e...
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作者：Neufeld, Leonie
作者单位：University of Bielefeld
摘要：We study weighted sums of free identically distributed self-adjoint random variables with weights chosen randomly from the unit sphere and show that the Kolmogorov distance between the distribution of such a weighted sum and Wigner's semicircle law is of order n(-1/2) with high probability. Replacing the Kolmogorov distance by a weaker pseudometric, we obtain a rate of convergence of order n(-1), thus providing a free analog of the Klartag-Sodin result in classical probability theory. Moreover...
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作者：Gouezel, Sebastien; Rousseau, Jerome; Stadlbauer, Manuel
作者单位：Universite de Rennes; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universidade Federal da Bahia
摘要：We study the minimal distance between two orbit segments of length n, in a random dynamical system with sufficiently good mixing properties. This problem has already been solved in non-random dynamical system, and on average in random dynamical systems (the so-called annealed version of the problem): it is known that the asymptotic behavior for this question is given by a dimension-like quantity associated to the invariant measure, called correlation dimension (or R & eacute;nyi entropy). We s...
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作者：Kwasnicki, Mateusz
作者单位：Wroclaw University of Science & Technology
摘要：We prove a non-extinction result for Fleming-Viot-type systems of two particles with dynamics described by an arbitrary symmetric Hunt process under the assumption that the reference measure is finite. Additionally, we describe an invariant measure for the system, we discuss its ergodicity, and we prove that the reference measure is a stationary measure for the embedded Markov chain of positions of the surviving particle at successive branching times.
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作者：Boedihardjo, H.; Geng, X.
作者单位：University of Warwick; University of Melbourne
摘要：We show that the tail probability of the rough line integral f(0)(1) phi(X-t)dY(t), where (X, Y) is a 2D fractional Brownian motion with Hurst parameter H is an element of (1/4, 1/2) and phi is a C-b(infinity)-function satisfying a mild non-degeneracy condition on its derivative, cannot decay faster than a gamma-Weibull tail with any exponent gamma > 2H + 1. In particular, this produces a simple class of examples of differential equations driven by fBM, whose solutions fail to have Gaussian ta...
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作者：Huang, Jiaoyang; Mcswiggen, Colin
作者单位：University of Pennsylvania; New York University
摘要：This paper studies the asymptotic behavior of several central objects in Dunkl theory as the dimension of the underlying space grows large. Our starting point is the observation that a recent result from the random matrix theory literature implies a large deviations principle for the hydrodynamic limit of radial Dunkl processes. Using this fact, we prove a variational formula for the large-N asymptotics of generalized Bessel functions, as well as a large deviations principle for the more gener...
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作者：Bazaes, Rodrigo; Lammers, Isabel; Mukherjee, Chiranjib
作者单位：University of Munster
摘要：We construct and study properties of an infinite dimensional analog of Kahane's theory of Gaussian multiplicative chaos (Kahane in Ann Sci Math Quebec 9(2):105-150, 1985). Namely, if H-T (omega) is a random field defined w.r.t. space-time white noise (B) over dot and integrated w.r.t. Brownian paths in d >= 3, we consider the renormalized exponential mu(gamma),T, weighted w.r.t. the Wiener measure P-0(d omega). We construct the almost sure limit mu(gamma) = lim(T ->) (infinity) (mu gamma,T) in...
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作者：Grazieschi, P.; Matetski, K.; Weber, H.
作者单位：University of Bath; Michigan State University; University of Munster
摘要：We consider multiple stochastic integrals with respect to c & agrave;dl & agrave;g martingales, which approximate a cylindrical Wiener process. We define a chaos expansion, analogous to the case of multiple Wiener stochastic integrals, for these integrals and use it to show moment bounds. Key tools include an iteration of the Burkholder-Davis-Gundy inequality and a multi-scale decomposition similar to the one developed in Hairer and Quastel (Forum Math Pi 6:e3, 2018). Our method can be combine...
