# Questions tagged [mixing]

For questions about mixing in ergodic or probability theory.

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### Bounds on $K\left(\mathcal{L}(X_t,X_{t+s})||\mathcal{L}(X_t)\otimes \mathcal{L}(X_{t+s})\right)$ for a diffusion process $(X_t)_{t\geq 0}$?

There are many results giving bounds on the $\beta$-mixing coefficients for diffusion processes (see Proposition 1 in  for example). For an homogeneous and stationary process, it means upper ...
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### Equivalent definition of weakly mixing

I found this exercise online some time ago, unfortunately I cannot remember where. I fail to see how to prove one of the implications, maybe someone here has an idea. Let $(X, \mathcal{F}, \mu, T)$ ...
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### Central Limit Theorem with Mixing

I would like to apply a CLT theorem to a sequence of random variables $X_1,X_2,X_3, \ldots$ which are "almost independent". Specifically, I want to apply a CLT theorem that replaces the ...
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### Applications where the mixing time of a Markov chain is important

I was wondering what the importance of the mixing time of a Markov chain is. I understand that the mixing time is a measure of how long it takes the distribution to approach the stationary ...
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### Is $Y_t = f(X_t, \beta_t)$ $\rho$-mixing if $X_t$ and $\beta_t$ are independent and each of them are $\rho$-mixing?

There is a post which says $Y_t = f(X_t, \beta_t)$ is strong mixing if $X_t$ and $\beta_t$ are independent and both strong mixing. Does this statement have any reference (book or paper)? In addition ...
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### Is this probability distribution part of a wider class of distributions?

The Hyperbolic distribution is characterized by a single log change of coordinates on the PDF yielding a hyperbola. What if we impose a double log change of coordinates on the PDF yielding a hyperbola?...
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### Strong mixing of a function of strong mixing and convergence sequence

Suppose the process $X = \left\{X_{t}:t\in Z\right\}$ is strong mixing with the coefficient $\alpha(j) \rightarrow 0$ defined as \begin{equation} \alpha(j)=\sup_T\sup_{1\leq k\leq T-j}\sup\left\{\...
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### Random walk on a graph whose vertices are $k$-tuples of distinct $n$-bit strings

I would appreciate any pointer or help with the following random walk. Consider a $k\times n$ $0$-$1$ matrix A, in which the following operation is performed repeatedly: Two different columns $C_i$ ...
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### What exactly is the probabilistic/intuitive interpretation of weak mixing of all orders? Why $A_0,\varphi^{\ast n}A_1,\varphi^{\ast 2n}A_2...$?

$\newcommand{\dlim}{\operatorname{Dlim}}\newcommand{\fix}{\operatorname{fix}}\newcommand{\1}{\mathbf{1}}$I am asking about the interpretations of certain ergodic dynamics definitions, citing this text ...
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### Random walk of $k$ particles on a $n$-dimensional hypercube

I would appreciate your help with the following, if possible. Consider an $n$-dimensional hypercube where nodes are connected by an edge if they differ in a single bit. There are $k=poly(n)$ particles,...
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### Mixing by the cat map : proof?

Let $f : \mathbb{T}^2 \to \mathbb{T}^2$ be the "cat map" defined by $f(x,y)=(2x+y \mod 1,x+y\mod 1)$. How can I prove that $f$ is exponentially mixing on $\mathbb{T}^2$ endowed with the ...
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### Prove this $\phi$ mixing inequality

Am trying to understand Theorem 2.2 in Serfling (1968): Proposition. Let $(\Omega,\mathcal A,P)$ be a probability space and $\mathcal F$ a sub-$\sigma$-algebra of $\mathcal A$. Let $X$ be a real ...
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In Theorem 4.9 of the book Markov Chains and Mixing Times by Levin & Peres, there is a convergence theorem for ergodic Markov chains which states that there exist constants $C > 0$ and $\alpha \... 0 votes 1 answer 196 views ### Meaning of Exact Transformation and$K$-Automorphism in the context of Ergodic Theory/Mixing I am reading MAGIC$010$Ergodic Theory course. In this course's lecture$4$notes, it is mentioned that$1)$Let$T$be an exact transformation of the probability space$(X,B,μ)$.Then$T$is strong-... 1 vote 0 answers 57 views ### Triangle array multiplied by an indicator function is also strong mixing (proof verification) Say that a triangle array$\{x_{n,i}:1\leq i\leq n, n\geq 1\}$of random variables on some probability space$\{\Omega, \mathcal{F}, P\}$is strong mixing if the coefficients $$\alpha_n(j)=\sup_{1\leq ... 0 votes 1 answer 74 views ### What is the relationship between P(\theta a + (1-\theta)b > 1) and P(a > 1), P(b > 1)? Suppose random value c is a mixture of a and b$$c = \theta a + (1-\theta)b$$What is the relationship of P(c > 1) and P(a>1), P(b>1), is it the following equation?$$ P(c > 1) ... 1 vote 1 answer 146 views ### Bounding the expectation of random variables' product. Let$\{X_i\}$be a sequence of strongly mixing random variables, not necessarily (strict) stationary. Assume that$E\lvert X_i\rvert^4\leq C<\infty$and that there is$0<a<1$such that the ... 1 vote 1 answer 270 views ### Lower bound of mixing time time of two glued complete graphs I am reading the "Markov Chains and mixing time 1st edition" by Levin and Peres, I got stuck on exercise$6.7$and have a hard time to understand its solution. Here is the description direct from ... 0 votes 0 answers 42 views ### Mixing time of combined Markov Chains Let$P_1,...,P_k$denote the transition matrices of$k$irreducible, aperiodic Markov Chains on finite state spaces$S_1,...,S_k$all of them being disjoint. Hence, there are stationary distributions$...
The mixing time $t_{mix}$ of a $\beta$ biased random walk on a line segment $\{0,1,\cdots n\}$ is given by $\beta^{-1}n+O(\sqrt{n})$. I am not interested in the proof but an understanding of the ...
Suppose $(X_i)_{i\in\mathbb{Z}}$ is a strongly mixing, strictly stationary sequence of random variables. If we assume $\mathbb{E}[|X_i|^{2+\epsilon}]<\infty$ for some $\epsilon>0$ and if we have ...