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1.

Curien, Nicolas

et al.

Ecole Normale Super, Dept Math & Applicat, F-75230 Paris 05, France..

Konstantopoulos, Takis

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory. Uppsala Univ, Dept Math, S-75106 Uppsala, Sweden..

Iterating Brownian Motions, Ad Libitum2014In: Journal of theoretical probability, ISSN 0894-9840, E-ISSN 1572-9230, Vol. 27, no 2, p. 433-448Article in journal (Refereed)

Abstract [en]

Let B-1, B-2, aEuro broken vertical bar be independent one-dimensional Brownian motions parameterized by the whole real line such that B (i) (0)=0 for every ia parts per thousand yen1. We consider the nth iterated Brownian motion W (n) (t)=B (n) (B (n-1)(a <-(B (2)(B (1)(t)))a <-)). Although the sequence of processes (W (n) ) (na parts per thousand yen1) does not converge in a functional sense, we prove that the finite-dimensional marginals converge. As a consequence, we deduce that the random occupation measures of W (n) converge to a random probability measure mu (a). We then prove that mu (a) almost surely has a continuous density which should be thought of as the local time process of the infinite iteration W (a) of independent Brownian motions. We also prove that the collection of random variables (W (a)(t),taa"ea-{0}) is exchangeable with directing measure mu(infinity).

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.

We consider a stochastic directed graph on the integers whereby a directed edge between i and a larger integer j exists with probability p(j-i) depending solely on the distance between the two integers. Under broad conditions, we identify a regenerative structure that enables us to prove limit theorems for the maximal path length in a long chunk of the graph. The model is an extension of a special case of graphs studied in [Markov Process. Related Fields 9 (2003) 413-468]. We then consider a similar type of graph but on the "slab" Z x I, where I is a finite partially ordered set. We extend the techniques introduced in the first part of the paper to obtain a central limit theorem for the longest path. When I is linearly ordered, the limiting distribution can be seen to be that of the largest eigenvalue of a |I| x |I| random matrix in the Gaussian unitary ensemble (GUE).

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.

Konstantopoulos, Takis

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.

Tucker, Warwick

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.

In this paper, we analyze the stability properties of a system of ordinary differential equations describing the thermodynamic limit of a microscopic and stochastic model for file sharing in a peer-to-peer network introduced by Kesidis et al. We show, under certain assumptions, that this BitTorrent-like system has a unique locally attracting equilibrium point which is also computed explicitly. Local and global stability are also shown.

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.

Konstantopoulos, Takis

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.

Yuan, Linglong

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory. Johannes Gutenberg Univ Mainz, Inst Math, Staudingerweg 9, D-55099 Mainz, Germany..

A random vector X = (X1, ... ,X-n) with the X-i taking values in an arbitrary measurable space (S, Sp) is exchangeable if its law is the same as that of (X-sigma(1), ... ,X-sigma(n)) for any permutation a. We give an alternative and shorter proof of the representation result (Jaynes [6] and Kerns and Szekely [9]) stating that the law of X is a mixture of product probability measures with respect to a signed mixing measure. The result is "finitistic" in nature meaning that it is a matter of linear algebra for finite S. The passing from finite S to an arbitrary one may pose some measure-theoretic difficulties which are avoided by our proof. The mixing signed measure is not unique (examples are given), but we pay more attention to the one constructed in the proof ("canonical mixing measure") by pointing out some of its characteristics. The mixing measure is, in general, defined on the space of probability measures on S; but for S =, one can choose a mixing measure on R-n.

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.

We prove the Cauchy-Binet determinantal formula using multilinear algebra by first generalizing it to an identity not involving determinants. By extending the formula to abstract Hilbert spaces we obtain, as a corollary, a generalization of the classical Parseval identity.

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.

Burke's theorem is a well-known fundamental result in queueing theory, stating that a stationary M/M/1 queue has a departure process that is identical in law to the arrival process and, moreover, for each time t, the following three random objects are independent: the queue length at time t, the arrival process after t and the departure process before t. Burke's theorem also holds for a stationary Brownian queue. In particular, it implies that a certain "complicated" functional derived from two independent Brownian motions is also a Brownian motion. The aim of this overview paper is to present an independent complete explanation of this phenomenon.

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.

Gabrysch, Katja

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.

We consider a directed graph on the 2-dimensional integer lattice, placing a directed edge from vertex (i(1), i(2)) to (j(1), j(2)), whenever i(1) <= j(1), i(2) <= j(2), with probability p, independently for each such pair of vertices. Let L-n,L-m denote the maximum length of all paths contained in an n x m rectangle. We show that there is a positive exponent a, such that, if m/n(a) -> 1, as n -> infinity, then a properly centered/rescaled version of L-n,L-m converges weakly to the Tracy-Widom distribution. A generalization to graphs with non-constant probabilities is also discussed.

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.

The longest stretch L(n) of consecutive heads in n independent and identically distributed coin tosses is seen from the prism of large deviations. We first establish precise asymptotics for the moment generating function of L(n) and then show that there are precisely two large deviation principles, one concerning the behavior of the distribution of L(n) near its nominal value log(1/p) n and one away from it. We discuss applications to inference and to logarithmic asymptotics of functionals of L(n).

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.

Trinajstić, Katja

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.

We consider a directed graph on the 2-dimensional integer lattice, placing a directed edge from vertex (i_{1},i_{2}) to (j_{1},j_{2}), whenever i_{1} ≤ j_{1}, i_{2} ≤ j_{2}, with probability p, independently for each such pair of vertices. Let L_{n,m} denote the maximum length of all paths contained in an n×m rectangle. We show that there is a positive exponent a, such that, if m/n^{a}→1, as n→∞, then a properly centered/rescaled version of L_{n,m} converges weakly to the Tracy-Widom distribution. A generalization to graphs with non-constant probabilities is also discussed.