For questions about mixing in ergodic or probability theory.

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Math Basis for Partial Fluid Change in Car Transmission

I have a fluid mixing problem with my car and I can't seem to find the answer: I have to change my transmission oil. The transmission has a 7 liter total capacity, but due to the torque converter, I ...
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0answers
18 views

Exponentially fast decay of alpha-mixing rates for irreducible, aperiodic finite, Markov chains

Let $(X_n)_{n \in \mathbb N}$ be a stationary, aperiodic, irreducible, finite state space Markov chain. Define the $\alpha$-mixing coefficient as: $$\alpha(n) = \sup \{\vert \Pr(A \cap B) - ...
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56 views

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} ...
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68 views

Finding the mixture of two gamma distributions

I am going through the theory behind mixing distributions, and I came across a portion in an essay where it talks about mixing two gamma distributions, it does not show how to get the resulting ...
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0answers
29 views

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 ...
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1answer
65 views

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 ...
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48 views

Mixing time for metropolis chain on graph coloring

I'm reading the Markov Chains and Mixing Times by David Levin et al.. In section 5.4 page 71 a proof is given for a bound of mixing time for the Metropolis Chain on graph coloring. In the proof, such ...
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1answer
72 views

Mixed Distributions - Expectation and Variance

A bike has probability of breaking down $p$, on any given day. The repair cost of the bike, whenever it breaks down, is distributed as a Gamma random variable with shape $\alpha$ and rate ...
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158 views

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; ...
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36 views

Calculate the mixing time of a continuous time markov chain

I have Markov Rate Matrix Q for a continuous time Markov chain, that is irreducible. I would like to calculate the mixing time of the matrix - how can I do so? Note that the methods in Markov Chains ...
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1answer
80 views

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 $$ 1)\qquad\lim_{t\rightarrow\infty} \mu(\phi^{-t}(A)\cap B)=\mu(A)\mu(B); $$ $$ ...
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72 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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22 views

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
30 views

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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1answer
53 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
104 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
40 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
61 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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29 views

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
89 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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66 views

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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105 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
78 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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43 views

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
369 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
204 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
240 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 ...
5
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1answer
120 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
158 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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354 views

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 ...