For questions about mixing in ergodic or probability theory.

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Alpha mixing property of a $\mathbb{R}^d$ valued Stochastic Process

In statistics and probability literature, a strictly stationary stochastic process $\{X_t\}\in\mathbb{R}$ is called $\alpha$-mixing if $\alpha(n)=\sup_{A\in\mathcal{F}_{-\infty}^{0}, ...
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Mixing System and density argument

A Mixing system is defined as a dynamical system $(\Omega,\phi^t,\mu)$ for which the following relations holds $$ 1)\qquad\lim_{t\rightarrow\infty} \mu(\phi^{-t}(A)\cap B)=\mu(A)\mu(B); $$ $$ ...
3
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1answer
48 views

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 ...
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Batch Linear Unmixing

Linear unmixing means to solve a set of linear equations to get the proportions of basic elements in the final composit. It is a linear mixture. If we have $f$ features and $m$ basic elements: ...
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0answers
35 views

Geometric ergodicity and mixing - stationary case

I have this theorem: The Markov Chain {$X_n$} is stationary and geometrically ergodic if and only if {$X_n$} is stationary and absolutely regular with $\beta_n=O(\gamma^n)$ for some $\gamma\in ...
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Markov chains mixing time

Informally, the mixing time of a Markov chain is the time it takes to reach “nearly uniform” distribution from any arbitrary starting distribution. What does it mean by nearly uniform? I hope some one ...
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31 views

A question on Bernoulli measures and mixing

this is a question on ergodic theory. Suppose I have an integer $N \geq2$ and a probability space $(\sum^{+} , B, \mu_{p})$, where $\mu_{p}$ is the Bernouilli measure with respect to probability ...
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1answer
63 views

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. ...
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1answer
27 views

mixing conditions for processes $(X_n, n\in \mathbb{N})$?

I hope this is not off topic, but I'm working with a process $(X_n, n\in \mathbb{N})$ and there's a theorem I want to use that is valid for certain strong mixing processes. The thing is, the usual ...
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1answer
51 views

DE Mixture Problem

Suppose a tank with a total capacity of 60 Gallons is currently only half full of a solution of water with 2% bleach concentration. At time t=0 water with a bleach concentration of 7% is pumped in at ...
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0answers
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generating locally random permutations

I have an intuitive notion of 'local randomness' that I am trying to make precise and understandable, and I am running into a bunch of problems. A quick web search failed to find anything relevant in ...
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1answer
79 views

If $X_i$ be a $\alpha$-mixing séquence, what about $X_i^2$?

Let $(X_i)_{i\in \mathbb{Z}}$ be an $\alpha$-mixing sequence of random variable. Is the sequence $(X_i^2)_{i\in\mathbb Z}$ also an $\alpha$-mixing sequence?
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1answer
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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 ...
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0answers
51 views

Mixture of discrete and continuous distributions

I have a question about how to find the pdf of a random variable $Z$ which is a sum and product of continuous AND discrete random variables. More specifically, I know that $X$ is Bernoulli($p$) and I ...
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1answer
70 views

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 ...
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1answer
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Need help with the following:

Proof or counterexample: a) if $T$ is ergodic, then $T^2$ is ergodic, b) if $T$ is strong mixing, then $T^2$ is strong mixing. Thank you.
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1answer
242 views

Ergodic Theory (Weak Mixing)

If $T$ is weak mixing then we know that $T\times T \times \ldots \times T$ is also weak mixing. Does anyone know if this is true for $T\times T \times \ldots$?
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1answer
119 views

Skewness of mixture density

I have the mixture density of two normal distributions: \begin{align} f(l)=\pi \phi(l;\mu_1,\sigma^2_1)+(1-\pi)\phi(l;\mu_2,\sigma^2_2) \end{align} The skewness is given by \begin{align} ...
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1answer
159 views

difference between convolution of two densities and mixture density?

I am wondering about the difference of the convolution of two probability density functions and the mixture of those two. This is not the same right? But what is the difference and how can it be ...
4
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1answer
92 views

What is the relation between mixing (measure theory) and a map being topological mixing?

A map is said to be topogical mixing if given two sets $A$ and $B$ then there exists $N$ such that for all $n>N$ $f^n(A) \cap B$ is not empty On the other hand, a measure \mu is said to be ...
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1answer
119 views

Is the product of two mixing random variables also mixing?

Question: Assume $X_t$ and $Y_t$ are random variables from the same probability space adapted to the filtration $\mathcal{F}_{-\infty}, ..., \mathcal{F}_t, ..., \mathcal{F}_{\infty}$. If $X_t$ and ...
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2answers
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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 ...