Mixing rates for Brownian motion in a convex polyhedron

1990 ◽  
Vol 27 (2) ◽  
pp. 259-268 ◽  
Author(s):  
Peter Matthews
Keyword(s):  
Brownian Motion ◽  
Rate Of Convergence ◽  
Convex Polyhedron ◽  
Upper Bounds ◽  
Convergence In Distribution ◽  
Convex Polyhedral ◽  
Variation Distance ◽  
Mixing Rates

For Brownian motion on a convex polyhedral subset of a sphere or torus, the rate of convergence in distribution to uniformity is studied. The main result is a method to take a Markov coupling on the full sphere or torus and create a faster coupling on the convex polyhedral subset. Upper bounds on variation distance are computed, and applications are discussed.

Mixing rates for Brownian motion in a convex polyhedron

1990 ◽  
Vol 27 (02) ◽  
pp. 259-268
Author(s):  
Peter Matthews
Keyword(s):  
Brownian Motion ◽  
Rate Of Convergence ◽  
Convex Polyhedron ◽  
Upper Bounds ◽  
Convergence In Distribution ◽  
Convex Polyhedral ◽  
Variation Distance ◽  
Mixing Rates

For Brownian motion on a convex polyhedral subset of a sphere or torus, the rate of convergence in distribution to uniformity is studied. The main result is a method to take a Markov coupling on the full sphere or torus and create a faster coupling on the convex polyhedral subset. Upper bounds on variation distance are computed, and applications are discussed.

scholarly journals Mixing rates for potentials of non-summable variations

2021 ◽  
pp. 1-18
Author(s):  
CHRISTOPHE GALLESCO ◽  
DANIEL Y. TAKAHASHI
Keyword(s):  
Invariance Principle ◽  
Type Inequality ◽  
Upper Bounds ◽  
Decay Of Correlations ◽  
Relaxation Rates ◽  
Weak Invariance Principle ◽  
Weak Invariance ◽  
Coupling Inequality ◽  
Mixing Rates

Abstract Mixing rates, relaxation rates, and decay of correlations for dynamics defined by potentials with summable variations are well understood, but little is known for non-summable variations. This paper exhibits upper bounds for these quantities for dynamics defined by potentials with square-summable variations. We obtain these bounds as corollaries of a new block coupling inequality between pairs of dynamics starting with different histories. As applications of our results, we prove a new weak invariance principle and a Hoeffding-type inequality.

scholarly journals Self-exciting multifractional processes

2021 ◽  
Vol 58 (1) ◽  
pp. 22-41
Author(s):  
Fabian A. Harang ◽  
Marc Lagunas-Merino ◽  
Salvador Ortiz-Latorre
Keyword(s):  
Stochastic Process ◽  
Brownian Motion ◽  
Rate Of Convergence ◽  
Existence And Uniqueness ◽  
Volterra Equation ◽  
The Self ◽  
The Past ◽  
Stochastic Volterra Equation ◽  
Multifractional Brownian Motion ◽  
Self Organizing

AbstractWe propose a new multifractional stochastic process which allows for self-exciting behavior, similar to what can be seen for example in earthquakes and other self-organizing phenomena. The process can be seen as an extension of a multifractional Brownian motion, where the Hurst function is dependent on the past of the process. We define this by means of a stochastic Volterra equation, and we prove existence and uniqueness of this equation, as well as giving bounds on the p-order moments, for all $p\geq1$. We show convergence of an Euler–Maruyama scheme for the process, and also give the rate of convergence, which is dependent on the self-exciting dynamics of the process. Moreover, we discuss various applications of this process, and give examples of different functions to model self-exciting behavior.

On the computability of a construction of Brownian motion

2013 ◽  
Vol 23 (6) ◽  
pp. 1257-1265 ◽  
Author(s):  
GEORGE DAVIE ◽  
WILLEM L. FOUCHÉ
Keyword(s):  
Brownian Motion ◽  
Lower Bound ◽  
Rate Of Convergence ◽  
Upper Bound ◽  
Binary Sequence ◽  
Computable Analysis

We examine a construction due to Fouché in which a Brownian motion is constructed from an algorithmically random infinite binary sequence. We show that although the construction is provably not computable in the sense of computable analysis, a lower bound for the rate of convergence is computable in any upper bound for the compressibilty of the sequence, making the construction layerwise computable.

Poisson approximation of multivariate Poisson mixtures

2003 ◽  
Vol 40 (02) ◽  
pp. 376-390 ◽  
Author(s):  
Bero Roos
Keyword(s):  
Total Variation ◽  
Poisson Approximation ◽  
Upper Bounds ◽  
Risk Theory ◽  
Total Variation Distance ◽  
Poisson Mixtures ◽  
Poisson Distributions ◽  
Variation Distance

We show how good multivariate Poisson mixtures can be approximated by multivariate Poisson distributions and related finite signed measures. Upper bounds for the total variation distance with applications to risk theory and generalized negative multinomial distributions are given. Furthermore, it turns out that the ideas used in this paper also lead to improvements in the Poisson approximation of generalized multinomial distributions.

On Simulating Wiener Integrals and Their Expectations

1991 ◽  
Vol 5 (1) ◽  
pp. 101-112 ◽  
Author(s):  
A. Korzeniowski ◽  
D.L. Hawkins
Keyword(s):  
Brownian Motion ◽  
Heat Equation ◽  
Rate Of Convergence ◽  
Approximation Scheme ◽  
Numerical Results

An approximation scheme for evaluating Wiener integrals by simulating Brownian motion is studied. The rate of convergence and numerical results are given, including an application to the heat equation by using the Feynman-Kac formula.

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