Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [mixing]

For questions about mixing in ergodic or probability theory.

In ergodic theory, there are two main notions of mixing: weak and strong mixing. They give how far is $\mu(A\cap T^{—n}B)$ from $\mu(A)\mu(B)$ if $(X,\mathcal,\mu,T)$ is a dynamical system.

In probability theory, mixing conditions are used to measure dependence between random variables.

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Sigma-algebra generated by a set of random variables

I know from standard textbooks that "Given the measurable functions $X_i:(\Omega,\mathcal{F})\rightarrow(\Omega_i,\mathcal{A}_i)$, the $\sigma$-algebra generated by a set of random variables $(X_i; i\in I)$ is given…
John
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An equivalent condition for strong-mixing

For a measure-preserving (finite) system $(X,\mathcal{B},\mu,T)$, is it correct that the following are equivalent? For every $A,B\in\mathcal{B}$ , $\displaystyle\lim_{n\rightarrow\infty}\mu(A\cap T^{-n}B)=\mu(A)\mu(B)$. For every…
user25640
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What is the relation between mixing (measure theory) and a map being topological mixing?

A map is said to be topological mixing if given two sets $A$ and $B$ then there exists $N$ such that for all $n>N$ $$f^n(A) \cap B \neq \emptyset.$$ On the other hand, a measure $\mu$ is said to be mixing for a map $f$ if $$\lim_{n \to \infty} \mu…
Zhör
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Intuitive explanation of the spectral gap in context of Markov Chain Monte Carlo (MCMC)

I'm learning about Markov Chains Monte Carlo methods and mixing times, and could use some help understanding the concept of the Spectral Gap and why / how it relates to the mixing times. Thus far what I have learned is this: given an ergodic…
user3280193
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Baker's map is ergodic

Define Baker's map by \begin{align} f(x,y) = \begin{cases} (2x,y/2) & \mbox{ if } (x,y)\in[0,1/2]\times[0,1] \\ (2x-1,y/2+1/2) & \mbox{ if } (x,y)\in[1/2,1]\times[0,1] \\ \end{cases} \end{align} I already proved that $f$ is invariant with respect…
Felipe Pérez
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A question on ergodic theory: topological mixing and invariant measures

This is a question on dynamical systems. Suppose I have a compact metric space $X$, with $([0,1], B, \mu)$ a probability space, with $B$ a (Borel) sigma algebra, and $\mu$ the probability measure. Suppose also that $\mu(A) > 0$ for any nonempty set…
Jack 33
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Definition of strong mixing and definition of measure-preserving

I have a few questions regarding measure-preserving dynamical systems $(X,\mathcal{A},\mu,T)$. 1) The definition of measure preserving is always stated as $$\mu(T^{-1}B)=\mu(B),$$ for all $B$. I believe this neither implies nor is implied by the…
angryavian
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Equivalence of the definitions of exactness and mixing

Let $f:X \to X$ be a continuous map, where $X$ is a compact metric space. We say that $f$ is expanding if there are constants $\lambda >1$ and $\delta_0 > 0$ such that, for all $x, y\in X$, $d(f(x), f(y)) \ge \lambda d(x, y)$ whenever $d(x, y) \le…
Mrcrg
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Strongly mixing uniquely ergodic dynamical system

I'm looking for an example of a dynamical system which is both (measure-theoretically) strongly mixing and uniquely ergodic. I've looked around and found lots of discussion of systems which are uniquely ergodic but not strongly mixing, but which…
Henry
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It Suffices to Check Mixing on an Algebra

Let $X, \mathcal{A}, \mu$ be a probability space and $T: X \rightarrow X$ a measure preserving measurable map (i.e. $\mu (T^{-1}(A)) = \mu (A)$ for all $A \in \mathcal{A}$). We say $T$ is mixing for sets $A, B \in \mathcal{A}$ if…
Mr. Cooperman
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Equivalent condition for mixing

Let $(X,\mathscr{A},\mu)$ be a measurable space and $T$ measure preserving map and define: $T$ is mixing iff $\lim_{n\to\infty}\mu(A\cap T^{-n}B)=\mu(A)\mu(B)$ for all $A,B\in\mathscr{A}$. How to prove that mixing property is equivalent…
alans
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Spectral gaps of common graphs

I'm looking for the spectral gap of common graphs (alternatively, the mixing time of a (lazy) random walk on these graphs). Asymptotic values are fine. Assume that every node has a sufficient number of self-loops. (For regular graphs at least d…
Xarph
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If $f^n$ is mixing then $f$ is mixing?

Let $(X,\mathcal{A},\mu)$ be a probability space and $f:X\to X$ be a measurable map that preserves $\mu$. Fix $n\in \mathbb{Z}^+$. It's not hard to see that $f$ ergodic does not necessarily imply $f^n$ ergodic. For example, take…
Bruno Stonek
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Bounding the Laplace transform (MGF) of a sum of dependent Bernoulli variables.

Consider a sequence $(X_k^{(n)})_{k\leqslant n}$ of nonnegative measurable functions from a probability space such that $\mathbb{E}X_k^{(n)}=p_n$ and $\|X_k^{(n)}\|_\infty\leqslant 1$. Suppose that $np_n\to 0$ as $n\to\infty$ and additionally assume…
Functional enthusiast
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A Talagrand inequality for the supremum of partial sums over function classes under dependence. (Reference request)

As a consequence to the Talagrand concentration inequality, it is well known that for a measurable space $(S,\mathcal{S})$ and an i.i.d. sample $X_1,...,X_n$ of $S$-valued random variables, if $\mathcal{F}$ is a countable set of measurable functions…
Daan
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