-
作者：Samorodnitsky, Gennady; Shen, Yi
作者单位：Cornell University
摘要：It is, perhaps, surprising that the location of the unique supremum of a stationary process on an interval can fail to be uniformly distributed over that interval. We show that this distribution is absolutely continuous in the interior of the interval and describe very specific conditions the density has to satisfy. We establish universal upper bounds on the density and demonstrate their optimality.
-
作者：Cass, Thomas; Litterer, Christian; Lyons, Terry
作者单位：Imperial College London; University of Oxford
摘要：We derive explicit tail-estimates for the Jacobian of the solution flow for stochastic differential equations driven by Gaussian rough paths. In particular, we deduce that the Jacobian has finite moments of all order for a wide class of Gaussian process including fractional Brownian motion with Hurst parameter H > 1/4. We remark on the relevance of such estimates to a number of significant open problems.
-
作者：Taylor, Jonathan E.; Vadlamani, Sreekar
作者单位：Stanford University; Tata Institute of Fundamental Research (TIFR); TIFR Centre for Applicable Mathematics (CAM), Bengaluru
摘要：In this work we consider infinite dimensional extensions of some finite dimensional Gaussian geometric functionals called the Gaussian Minkowsld functionals. These functionals appear as coefficients in the probability content of a tube around a convex set D subset of R-k under the standard Gaussian law N(0, I-kxk). Using these infinite dimensional extensions, we consider geometric properties of some smooth random fields in the spirit of [Random Fields and Geometry (2007) Springer] that can be ...
-
作者：Kuelbs, James; Kurtz, Thomas; Zinn, Joel
作者单位：University of Wisconsin System; University of Wisconsin Madison; Texas A&M University System; Texas A&M University College Station
摘要：For stochastic processes {X-t : t is an element of E}, we establish sufficient conditions for the empirical process based on {I-Xt <= y - Pr(X-t <= y) : t is an element of E, y is an element of R} to satisfy the CLT uniformly in t is an element of E, y is an element of R. Corollaries of our main result include examples of classical processes where the CLT holds, and we also show that it fails for Brownian motion tied down at zero and E = [0, 1].
-
作者：Ding, Jian
作者单位：Stanford University
摘要：For a complete graph of size n, assign each edge an i.i.d. exponential variable with mean n. For lambda > 0, consider the length of the longest path whose average weight is at most lambda. It was shown by Aldous [Combin. Probab. Comput. 7 (1998) 1-10] that the length is of order log n for lambda < 1/e and of order n for lambda > 1/e. Aldous [Open problems (2003) Preprint] posed the question on detailed behavior at and near criticality 1/e. In particular, Aldous asked whether there exist scalin...
-
作者：Dedecker, Jerome; Merlevede, Florence; Rio, Emmanuel
作者单位：Universite Paris Cite; Universite Gustave-Eiffel; Universite Paris-Est-Creteil-Val-de-Marne (UPEC); Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite Paris Saclay
摘要：We prove a strong approximation result for the empirical process associated to a stationary sequence of real-valued random variables, under dependence conditions involving only indicators of half lines. This strong approximation result also holds for the empirical process associated to iterates of expanding maps with a neutral fixed point at zero, as soon as the correlations decrease more rapidly than n(-1-delta) for some positive delta. This shows that our conditions are in some sense optimal.
-
作者：Erdos, Laszlo; Knowles, Antti; Yau, Horng-Tzer; Yin, Jun
作者单位：University of Munich; Harvard University; University of Wisconsin System; University of Wisconsin Madison
摘要：We consider the ensemble of adjacency matrices of Erdos-Renyi random graphs, that is, graphs on N vertices where every edge is chosen independently and with probability p equivalent to p(N). We rescale the matrix so that its bulk eigenvalues are of order one. We prove that, as long as pN -> infinity (with a speed at least logarithmic in N), the density of eigenvalues of the Erdos-Renyi ensemble is given by the Wigner semicircle law for spectral windows of length larger than N-1 (up to logarith...
-
作者：Hu, Yaozhong; Jolis, Maria; Tindel, Samy
作者单位：University of Kansas; Autonomous University of Barcelona; Universite de Lorraine
摘要：In this article, we derive a Stratonovich and Skorohod-type change of variables formula for a multidimensional Gaussian process with low Holder regularity gamma (typically gamma <= 1/4). To this aim, we combine tools from rough paths theory and stochastic analysis.
-
作者：Kwasnicki, Mateusz; Malecki, Jacek; Ryznar, Michal
作者单位：Wroclaw University of Science & Technology; Polish Academy of Sciences; Institute of Mathematics of the Polish Academy of Sciences; Universite d'Angers
摘要：In this paper we study the supremum functional M-t = sup(0 <= s <= t) X-s, where X-t, t >= 0, is a one-dimensional Levy process. Under very mild assumptions we provide a simple, uniform estimate of the cumulative distribution function of Mt. In the symmetric case we find an integral representation of the Laplace transform of the distribution of M-t if the Levy-Khintchin exponent of the process increases on (0, infinity).
-
作者：Panchenko, Dmitry
作者单位：Texas A&M University System; Texas A&M University College Station
摘要：In many spin glass models, due to the symmetry among sites, any limiting joint distribution of spins under the annealed Gibbs measure admits the Aldous-Hoover representation encoded by a function sigma : [0, 1](4) -> {-1, +1}, and one can think of this function as a generic functional order parameter of the model. In a class of diluted models, and in the Sherrington -Kirkpatrick model, we introduce novel perturbation's of the Hamiltonian that yield certain invariance and self-consistency equat...
