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作者：Burdzy, Krzysztof; Chen, Zhen-Qing
作者单位：University of Washington; University of Washington Seattle
摘要：In this paper, we show that reflecting Brownian motion in any bounded domain D can be approximated, as k -> infinity, by simple random walks on maximal connected subsets of (2(-k)Z(d)) boolean AND D whose filled-in interiors are inside of D.
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作者：Deroin, B.; Kleptsyn, V.; Navas, A.; Parwani, K.
作者单位：Centre National de la Recherche Scientifique (CNRS); Universite Paris Saclay; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite de Rennes; Universidad de Santiago de Chile; Eastern Illinois University
摘要：We study symmetric random walks on finitely generated groups of orientation-preserving homeomorphisms of the real line. We establish an oscillation property for the induced Markov chain on the line that implies a weak form of recurrence. Except for a few special cases, which can be treated separately, we prove a property of global stability at a finite distance: roughly speaking, there exists a compact interval such that any two trajectories get closer and closer whenever one of them returns t...
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作者：Duminil-Copin, H.; Van Enter, A. C. D.
作者单位：University of Geneva; University of Groningen
摘要：Bootstrap percolation models have been extensively studied during the two past decades. In this article, we study the following anisotropic boot-strap percolation model: the neighborhood of a point (m, n) is the set {(m + 2, n), (m + 1, n), (m, n + 1), (m - 1, n), (m - 2, n), (m, n - 1)}. At time 0, sites are occupied with probability p. At each time step, sites that are occupied remain occupied, while sites that are not occupied become occupied if and only if three of more sites in their neig...
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作者：Kabluchko, Zakhar; Zaporozhets, Dmitry
作者单位：Ulm University; Russian Academy of Sciences
摘要：It has been shown by Ibragimov and Zaporozhets [In Prokhorov and Contemporary Probability Theory (2013) Springer] that the complex roots of a random polynomial G(n)(z) = Sigma(n)(k=0) xi(k)z(k) with i.i.d. coefficients xi(0), ... , xi(n) concentrate a.s. near the unit circle as n -> infinity if and only if E log(+)vertical bar xi(0)vertical bar < infinity. We study the transition from concentration to deconcentration of roots by considering coefficients with tails behaving like L(log vertical ...
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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.
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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.
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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 ...
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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].
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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...
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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.
