-
作者：Rohde, Angelika; Duembgen, Lutz
作者单位：University of Hamburg; University of Bern
摘要：In the setting of high-dimensional linear models with Gaussian noise, we investigate the possibility of confidence statements connected to model selection. Although there exist numerous procedures for adaptive (point) estimation, the construction of adaptive confidence regions is severely limited (cf. Li in Ann Stat 17:1001-1008, 1989). The present paper sheds new light on this gap. We develop exact and adaptive confidence regions for the best approximating model in terms of risk. One of our c...
-
作者：Hairer, Martin; Weber, Hendrik
作者单位：University of Warwick
-
作者：Meckes, Elizabeth S.; Meckes, Mark W.
作者单位：University System of Ohio; Case Western Reserve University
摘要：The topic of this paper is the typical behavior of the spectral measures of large random matrices drawn from several ensembles of interest, including in particular matrices drawn from Haar measure on the classical Lie groups, random compressions of random Hermitian matrices, and the so-called random sum of two independent random matrices. In each case, we estimate the expected Wasserstein distance from the empirical spectral measure to a deterministic reference measure, and prove a concentrati...
-
作者：Schramm, Oded; Sheffield, Scott
作者单位：Massachusetts Institute of Technology (MIT)
摘要：Consider an instance of the Gaussian free field on a simply connected planar domain with boundary conditions on one boundary arc and on the complementary arc, where is the special constant . We argue that even though is defined only as a random distribution, and not as a function, it has a well-defined zero level line connecting the endpoints of these arcs, and the law of is . We construct in two ways: as the limit of the chordal zero contour lines of the projections of onto certain spaces of ...
-
作者：van de Geer, Sara; Lederer, Johannes
作者单位：Swiss Federal Institutes of Technology Domain; ETH Zurich
摘要：We introduce two new concepts designed for the study of empirical processes. First, we introduce a new Orlicz norm which we call the Bernstein-Orlicz norm. This new norm interpolates sub-Gaussian and sub-exponential tail behavior. In particular, we show how this norm can be used to simplify the derivation of deviation inequalities for suprema of collections of random variables. Secondly, we introduce chaining and generic chaining along a tree. These simplify the well-known concepts of chaining...
-
作者：Cai, T. Tony; Ren, Zhao; Zhou, Harrison H.
作者单位：University of Pennsylvania; Yale University
摘要：Toeplitz covariance matrices are used in the analysis of stationary stochastic processes and a wide range of applications including radar imaging, target detection, speech recognition, and communications systems. In this paper, we consider optimal estimation of large Toeplitz covariance matrices and establish the minimax rate of convergence for two commonly used parameter spaces under the spectral norm. The properties of the tapering and banding estimators are studied in detail and are used to...
-
作者：Peres, Yuval; Sinclair, Alistair; Sousi, Perla; Stauffer, Alexandre
作者单位：Microsoft; University of California System; University of California Berkeley; University of Cambridge
摘要：We consider the following dynamic Boolean model introduced by van den Berg et al. (Stoch. Process. Appl. 69:247-257, 1997). At time 0, let the nodes of the graph be a Poisson point process in with constant intensity and let each node move independently according to Brownian motion. At any time t, we put an edge between every pair of nodes whose distance is at most r. We study three fundamental problems in this model: detection (the time until a target point-fixed or moving-is within distance r...
-
作者：Tao, Terence
作者单位：University of California System; University of California Los Angeles
摘要：It is known that if one perturbs a large iid random matrix by a bounded rank error, then the majority of the eigenvalues will remain distributed according to the circular law. However, the bounded rank perturbation may also create one or more outlier eigenvalues. We show that if the perturbation is small, then the outlier eigenvalues are created next to the outlier eigenvalues of the bounded rank perturbation; but if the perturbation is large, then many more outliers can be created, and their ...
-
作者：Croydon, David A.
作者单位：University of Warwick
摘要：In this article, a localisation result is proved for the biased random walk on the range of a simple random walk in high dimensions (). This demonstrates that, unlike in the supercritical percolation setting, a slowdown effect occurs as soon as a non-trivial bias is introduced. The proof applies a decomposition of the underlying simple random walk path at its cut-times to relate the associated biased random walk to a one-dimensional random walk in a random environment in Sinai's regime. Via th...
-
作者：Mikosch, Thomas; Moser, Martin
作者单位：University of Copenhagen; Technical University of Munich; Technical University of Munich
摘要：We investigate the maximum increment of a random walk with heavy-tailed jump size distribution. Here heavy-tailedness is understood as regular variation of the finite-dimensional distributions. The jump sizes constitute a strictly stationary sequence. Using a continuous mapping argument acting on the point processes of the normalized jump sizes, we prove that the maximum increment of the random walk converges in distribution to a Fr,chet distributed random variable.
