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Probability Surveys
Probability Surveys
Limic, Vlada; Limić, Nedžad
The theory of equidistribution is about hundred years old, and has been developed primarily by number theorists and theoretical computer scientists. A motivated uninitiated peer could encounter difficulties perusing the literature, due to various synonyms and polysemes used by different schools. One purpose of this note is to provide a short introduction for probabilists. We proceed by recalling a perspective originating in a work of the second author from 2002. Using it, various new examples of completely uniformly distributed $\mathsf{mod}~1$ sequences, in the “metric” (meaning almost sure stochastic) sense, can be easily exhibited. In particular, we point out natural generalizations...
Duminil-Copin, Hugo; Gagnebin, Maxime; Harel, Matan; Manolescu, Ioan; Tassion, Vincent
In this paper, we review a few known facts on the coordinate Bethe ansatz. We present a detailed construction of the Bethe ansatz vector $\psi $ and energy $\Lambda $, which satisfy $V\psi =\Lambda \psi $, where $V$ is the transfer matrix of the six-vertex model on a finite square lattice with periodic boundary conditions for weights $a=b=1$ and $c>0$. We also show that the same vector $\psi $ satisfies $H\psi =E\psi $, where $H$ is the Hamiltonian of the XXZ model (which is the model for which the Bethe ansatz was first developed), with a value $E$ computed explicitly.
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Lawler, Gregory F.
Ferrari, Pablo A.
The convergence of the totally asymmetric simple exclusion process to the solution of the Burgers equation is a classical result. In his seminal 1981 paper, Herman Rost proved the convergence of the density fields and local equilibrium when the limiting solution of the equation is a rarefaction fan. An important tool of his proof is the subadditive ergodic theorem. We prove his results by showing how second class particles transport the rarefaction-fan solution, as characteristics do for the Burgers equation, avoiding subadditivity. Along the way we show laws of large numbers for tagged particles, fluxes and second class particles, and...
Ibragimov, Rustam; Lentzas, George
This paper focuses on the analysis of persistence properties of copula-based time series. We obtain theoretical results that demonstrate that Gaussian and Eyraud-Farlie-Gumbel-Morgenstern copulas always produce short memory stationary Markov processes. We further show via simulations that, in finite samples, stationary Markov processes, such as those generated by Clayton copulas, may exhibit a spurious long memory-like behavior on the level of copulas, as indicated by standard methods of inference and estimation for long memory time series. We also discuss applications of copula-based Markov processes to volatility modeling and the analysis of nonlinear dependence properties of returns in real financial markets...
Hammond, Alan
The Smoluchowski coagulation-diffusion PDE is a system of partial differential equations modelling the evolution in time of mass-bearing Brownian particles which are subject to short-range pairwise coagulation. This survey presents a fairly detailed exposition of the kinetic limit derivation of the Smoluchowski PDE from a microscopic model of many coagulating Brownian particles that was undertaken in [11]. It presents heuristic explanations of the form of the main theorem before discussing the proof, and presents key estimates in that proof using a novel probabilistic technique. The survey’s principal aim is an exposition of this kinetic limit derivation, but it also contains...
Mossel, Elchanan; Tamuz, Omer
Holmgren, Cecilia; Janson, Svante
This survey studies asymptotics of random fringe trees and extended fringe trees in random trees that can be constructed as family trees of a Crump–Mode–Jagers branching process, stopped at a suitable time. This includes random recursive trees, preferential attachment trees, fragmentation trees, binary search trees and (more generally) $m$-ary search trees, as well as some other classes of random trees.
¶ We begin with general results, mainly due to Aldous (1991) and Jagers and Nerman (1984). The general results are applied to fringe trees and extended fringe trees for several particular types of random trees, where the theory is developed in...
Ley, Christophe; Reinert, Gesine; Swan, Yvik
We propose a new general version of Stein’s method for univariate distributions. In particular we propose a canonical definition of the Stein operator of a probability distribution which is based on a linear difference or differential-type operator. The resulting Stein identity highlights the unifying theme behind the literature on Stein’s method (both for continuous and discrete distributions). Viewing the Stein operator as an operator acting on pairs of functions, we provide an extensive toolkit for distributional comparisons. Several abstract approximation theorems are provided. Our approach is illustrated for comparison of several pairs of distributions: normal vs normal, sums of independent...