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Category Archives: statistical laws
Law of large numbers for dependent but uncorrelated random variables
One of the fundamental results in probability theory is the strong law of large numbers, which was discussed in an earlier post under the guise of the Birkhoff ergodic theorem. Suppose we have a sequence of random variables which take … Continue reading
Equidistribution for random rotations
Two very different types of dynamical behaviour are illustrated by a pair of very wellknown examples on the circle: the doubling map and an irrational rotation. On the unit circle in , the doubling map is given by , while … Continue reading
Posted in ergodic theory, examples, random dynamics, statistical laws
Tagged space of invariant measures
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Central Limit Theorem for dynamical systems using martingales
This post is based on notes from Matt Nicol’s talk at the UH summer school in dynamical systems. The goal is to present the ideas behind a proof of the central limit theorem for dynamical systems using martingale approximations. 1. … Continue reading
Posted in ergodic theory, statistical laws, theorems
Tagged central limit theorem, martingales
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Spectral methods 3 – central limit theorem
With the previous post on convergence of random variables, the law of large numbers, and Birkhoff’s ergodic theorem as background, we return to the spectral methods discussed in the first two posts in this series. This post is based on … Continue reading
Laws of large numbers and Birkhoff’s ergodic theorem
In preparation for the next post on the central limit theorem, it’s worth recalling the fundamental results on convergence of the average of a sequence of random variables: the law of large numbers (both weak and strong), and its strengthening … Continue reading
Markov chains and mixing times (part 3)
The previous post introduced the idea of coupling for Markov chains as a method for estimating mixing times. Here we mention a particular example of a coupling that is often useful — this is the classical coupling, or Doeblin coupling, … Continue reading
Markov chains and mixing times (part 2 – coupling)
This week’s post continues last week’s discussion of Markov chains and mixing times, and introduces the idea of coupling as a method for estimating mixing times. We remark that some nice notes on the subject of coupling (and others) can … Continue reading
Markov chains and mixing times
Our seminar series is taking a hiatus from spectral methods for a couple weeks — these will return eventually, but in the meantime we’ll spend some time with the idea of coupling as a method for deriving statistical properties. In … Continue reading
Posted in examples, statistical laws
Tagged decay of correlations, Markov chains, PerronFrobenius, spectral methods
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Spectral methods in dynamics (part 2)
This is a continuation of the last post, which were notes from the first in a series of talks at the Houston dynamics seminar on spectral methods for transfer operators as tools to establish statistical properties of dynamical systems. This … Continue reading
Posted in ergodic theory, examples, statistical laws
Tagged interval maps, physical measures, spectral methods, transfer operator
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Spectral methods in dynamics
In the dynamics seminar here at Houston, we’re beginning a series of expository talks on statistical properties of dynamical systems. This week’s talk was given by Andrew Török and introduces some of the spectral methods for transfer operators that prove … Continue reading