# Questions tagged [mixing]

For questions about mixing in ergodic or probability theory.

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### When is an ergodic Ito diffusion also mixing

If $(X_t,t\geq0)$ denotes the strong solution to the Ito SDE $\mathrm{d}X_t=b(X_t)\mathrm{d}t+\sigma(X_t)\mathrm{d}W_t$ for $W_t$ the standard $d$-dimensional Brownian motion and $b,\sigma$ satisfying ...
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### KL Markov chain mixing result

Suppose $P$ is the transition matrix of a Markov chain with stationary distribution $\pi$. I am wondering what results can be said about how $P$ mixes distributions under the KL divergence? In ...
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### Product of coordinates of alpha mixing random vector.

I have the lemma that for any measurable function and $\alpha$-mixing process $\{x_t\}$, where $x_t$ can be a sequence of random vectors or just univariate random variables. Then $\{f(x_t)\}$, is also ...
• 343
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### Any hyperbolic automorphism of $\mathbb{T}^n$ is mixing.

We know a hyperbolic toral automorphism is defined to be an automorphism of $\mathbb{T}^n=\mathbb{R}^n/\mathbb{Z}^n$ induced by integer matrix in $\text{GL}(n,\mathbb{Z})$ which has no eigenvalue of ...
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### What Variance equation is this textbook using?

I'm doing a question on a Gamma/Poisson mix, which turns into a negative binomial. What I don't understand is the answer to the question - most of the answer makes sense to me (normal approximation ...
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### Expected node probability of given degree after finite random walk

Let $G=(V,E)$ be a connected, non-bipartite graph with $n$ nodes and $m$ edges. Consider a random walk of length $t$ on $G$ that starts at a uniform random node. The sequence of nodes in the random ...
1 vote
31 views

### Why is a completely random measure mixing?

Let $S_x$ denote an operator on $\mathcal{M}_{\chi}^{\#}$ by $S_x\xi(\cdot)=\xi(\cdot+x)$, where $\xi$ is a random measure. Here, $\mathcal{M}_{\chi}^{\#}$ refers to the space of boundedly finite ...
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### (Exponential) Mixing for Gauss map - going from cylinders to intervals

I'm trying to understand the proof of the mixing property of the Gauss map from the paper - 'Some metrical theorems in number theory' and I'm getting confused by the logic in a step. The Gauss map $T$ ...
51 views

### Modelling dependence under strong mixing conditions

Assume that the sequence $\{X_i\}_{i\in\mathbb N^*}$ of random variables on $(\Omega,\mathcal A,P)$ is strongly mixing. Denote by $\mathcal F_i^k=\sigma(X_t, i\leq t\leq k)$ a sigma-algebra generated. ...
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### Proof writing clarification.

Problem Let $X_n$ be a sequence of mixing random variables. If there is $a>1$ such that the strong mixing coefficient satisfies $$\alpha(s)\leq Cs^{-a}\quad (1)$$ ($C>0$ is a generic positive ...
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1 vote
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### Counting samples that are different in the given percentage of elements

Suppose there are n unique elements. A mix is a set of m elements pulled from the elements without replacement, so that there are $C^{m}_{n}$ possible mixes in total. Suppose one has all such mixes. ...
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### Mixing inequality in the paper "The absence of mixing in special flows over rearrangements of segments"

I can't verify an inequality in the paper: https://link.springer.com/content/pdf/10.1007/BF02110361.pdf The inequality is at the end of the second page in the proof of the second theorem and is shown ...
1 vote
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