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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 ...
Daan's user avatar
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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 ...
Tyler6's user avatar
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1 vote
2 answers
113 views

Markov Chain Transition Matrix eigenvalues - Textbook Exercise

I am trying to solve the following problem taken the book Markov Chains and Mixing Times 2nd edition (Exercise 12.2): Let $P$ be irreducible transition matrix, and suppose that A is a matrix with $0 \...
codeplay's user avatar
1 vote
0 answers
82 views

Simplest proof that exactness implies mixing

Let $f$ be a continuous map defined on a compact metric space $X$. Suppose that $f$ preserves the Borel probability measure $\mu$ and that, for every positive-measure set $A\subseteq X$, we have $$\...
Uagi's user avatar
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Is a random vector constructed of stacking alpha mixing random vectors also alpha mixing

If $X_t$ and $Y_t$ are both alpha mixing in the same sense for example decaying exponentially $\alpha(\ell) = O(\exp(-cn))$, $0 < c$. Then can we say that the stacked random vector process $(X_t, ...
Dylan Dijk's user avatar
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0 answers
22 views

Stuck on proof equidistribution on homogeneous space

I have been trying to understand the following proof but I don't fully understand the implicit last steps. From the replacement of the groups to the deducing of the theorems. I wondered if anyone ...
Jaero's user avatar
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1 answer
26 views

A question related to Lovasz Simonovits curve

I am trying to understand the Lovasz-Simonovits Theorem for random walks from this scribe note. For a graph $G$ with $2m$ edges, we have some edge measure $\rho$ and we order the edges according to ...
Sudipta Roy's user avatar
1 vote
1 answer
50 views

Help deriving this expression for mixing probabilities from Billingsley (1968)

I am wondering whether someone can help me derive this equation in Billingsley (1968), Equation (20.49)? Suppose $u_i$ and $v_j$ are integers with $u_1 \leq v_1 < u_2 \leq v_2 < \dots < u_r \...
KeynesCoeFen's user avatar
0 votes
1 answer
36 views

Calculating cold temp on mixing valve

I need a formula to calculate the cold temperature on a mixing valve system when given the hot temp, mixing percentage, and mixed temp. I can determine the hot and mixed temps with temp probes, but ...
Geo's user avatar
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191 views

Expanding endomorphism is mixing - proof details

When I read Introduction to Dynamical Systems by Brin and Stuck, I didn't understand one detail. This is in the proof showing that expanding endomorphism is mixing. I do not understand why when $n>...
FactorY's user avatar
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4 votes
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Is an AR(1) process with Bernoulli errors mixing or ergodic?

Before the $\text{AR}(1)$ model, first look at a simpler example $$y_t=\rho^t y_0+\epsilon_t$$ where $0<\rho<1$ and $\epsilon_t\overset{\text{i.i.d.}}{\sim} \text{Bernoulli} \left(\frac{1}{2} \...
Jack's user avatar
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1 answer
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Mixing property of a iid process

Let $(X_n)_{n\in \mathcal{N_0}}$ an i.i.d. sequence on a generic $E$, $\tau$ be the measure preserving shift operator and suppose there exist, for $A,B\in\mathcal{A}$,the product sigma algebra, $A^\...
Enrico's user avatar
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Properties of unitary representations for projective representations

I would like to know if there exist concepts of ergodicity and mixing properties for projective representations. If they do, do these properties exhibit similar characteristics to those observed in ...
aqwer's user avatar
  • 153
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1 answer
92 views

Weak law of large numbers, strong mixing

Let $\{X_n\}_{n\in \mathbb{N}}$ be a strong mixing sequence (no stationarity) with exponentially decaying mixing rate. Further assume that $X_n$ has uniformly bounded forth moments. Does $$\frac{1}{n}\...
statuser123's user avatar
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1 answer
60 views

How can I check that the following map is not topologically mixing?

Let me consider the matrix $$ M=\begin{bmatrix} 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 1 \end{bmatrix} $$ We consider the ...
user1294729's user avatar
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6 votes
1 answer
173 views

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(...
Mrcrg's user avatar
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1 vote
1 answer
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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 ...
Dylan Dijk's user avatar
2 votes
1 answer
368 views

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 ...
Pina Rinith's user avatar
1 vote
1 answer
39 views

Functions of uniform-mixing processes

Let $\{X_t\}_{t=1}^\infty$ be a stationary process which is uniform mixing with coefficients $\phi_r$ such that $\sum_{r=}^\infty\phi_r<\infty$. Consider the periodogram for $\omega\in[-\pi,\pi]$ $$...
wonderer's user avatar
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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 [1] for example). For an homogeneous and stationary process, it means upper ...
Zini's user avatar
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2 votes
1 answer
547 views

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 $(X, \mathcal{F}, \mu, T)$ ...
noam.szyfer's user avatar
  • 1,600
1 vote
1 answer
461 views

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 ...
Massimo's user avatar
  • 153
2 votes
1 answer
199 views

Are continuous mixtures of the gamma distribution identifiable with respect to the scale parameter?

Consider the two-parameter Gamma($\alpha$,$\beta$) distribution with PDF $$f(x|\alpha,\beta) = \frac{\beta^\alpha x^{\alpha - 1} \exp(-\beta x)}{\Gamma(\alpha)}, \quad x>0, \alpha>0, \beta>0,$...
BMG's user avatar
  • 103
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1 answer
280 views

Mixing time of discrete random walk on N-cycle

I would like to know the mixing time of a discrete random walk on an N-cycle which moves to the right or left with probability $\frac{1}{2}$. I read in a paper that this mixing time is $$M_\epsilon \...
Q.Ask's user avatar
  • 147
0 votes
2 answers
261 views

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 ...
Q.Ask's user avatar
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45 views

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 ...
Sabbir Ahmed's user avatar
1 vote
0 answers
70 views

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?...
zeta space's user avatar
4 votes
1 answer
273 views

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\{\...
Abdoul Haki's user avatar
2 votes
0 answers
29 views

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 $C_i$ ...
TDM's user avatar
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2 votes
1 answer
186 views

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 ...
FShrike's user avatar
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1 vote
0 answers
104 views

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,...
TDM's user avatar
  • 31
2 votes
0 answers
144 views

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 ...
Jacques Mardot's user avatar
1 vote
2 answers
190 views

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 ...
Alphie's user avatar
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3 votes
1 answer
97 views

How to show this $\phi$ mixing inequality?

Let $(\Omega,\mathcal A,P)$ be a probability space and $\mathcal F$ a sub-$\sigma$-algebra of $\mathcal A$. Am trying to show that $$ |P(A|\mathcal F)-P(A)|\leq \phi(\mathcal A,\mathcal F) \quad \quad ...
Alphie's user avatar
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1 vote
0 answers
44 views

Limit theorem for a conditional distribution on stationary mixing fields

For a project, I am looking into central limit theorems for stationary mixing random fields. I found many results for the asymptotic distribution of partial sums, such as that of Bolthausen (1982). ...
Seb's user avatar
  • 23
1 vote
1 answer
44 views

A continuous distribution that "mixes easily" with the uniform.

I want to consider a mixture of uniform distributions between $0$ and $t-U$ such that the $U$ is itself drawn from some mixing distribution. I want to pick a distribution such that the PDF (and ...
Rohit Pandey's user avatar
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4 votes
1 answer
350 views

Exponential convergence to equilibrium for finite continuous-time Markov chains

Setting: Consider a continuous-time Markov chain on a finite state space $\mathcal Z$ with irreducible (constant) generator matrix $L\in\mathbb R^{|\mathcal Z| \times |\mathcal Z|}$. Let $\rho_t\in\...
UPS's user avatar
  • 569
0 votes
1 answer
26 views

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 ...
Elie's user avatar
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2 votes
0 answers
105 views

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 ...
user134331092's user avatar
1 vote
0 answers
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 ...
Gerry T.'s user avatar
  • 119
3 votes
1 answer
83 views

(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 $T$ ...
Bim Binsella's user avatar
2 votes
0 answers
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. ...
Celine Harumi's user avatar
2 votes
0 answers
70 views

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 ...
Celine Harumi's user avatar
1 vote
1 answer
35 views

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. ...
Ivan K.'s user avatar
  • 113
2 votes
0 answers
64 views

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 ...
Clement Moreno's user avatar
1 vote
0 answers
33 views

Spatial covariance decomposition

$\{\mathbf{Z}_{\mathbf{i}}\}_{\mathbf{i} \in Z^{d}}$ is a strictly stationary process, following an isotropic short memory dependence with the covariance matrix given by $\big\{ C_{k,l}(\lVert \...
C2plus's user avatar
  • 11
2 votes
1 answer
127 views

Bounding a probability using a strong mixing property

Definition. Let $\{X_t, t>0\}$ be a sequence of real random variables on $(\Omega,\mathcal A,P)$ and denote by $\mathcal F_i^k=\sigma(X_t, i\leq t\leq k)$ the sigma-algebra generated. For any ...
Celine Harumi's user avatar
1 vote
1 answer
76 views

Mixing time of a lifted Markov Chain.

Let $(X_t)$ be a Markov chain and $(Y_t)$ a lifted chain of $(X_t)$, that is, the lifted chain splits each state in several states. $t_O$ denotes the mixing time of the original chain; $t_L$ denotes ...
Babado's user avatar
  • 1,316
0 votes
1 answer
161 views

Help with a covariance inequality

Let $(X_i)_{i\in\mathbb{N}}$ be a sequence of real-valued random variables. In a proof of a theorem, the author starts with $$\lvert Cov(X_i,X_j)\rvert\leq \int_0^{\alpha_{\lvert i-j\rvert}}[Q_i(u)]^2+...
Celine Harumi's user avatar
0 votes
1 answer
341 views

Covariance and indicator function relations

Let $\mathcal A$ and $\mathcal B$ be two (sub) sigma-algebras of some probability space. If $A\in\mathcal A$ and $B\in\mathcal B$, why the following equality holds: $$Cov(1_A-1_{A^c},1_B)=E((P(B\...
Celine Harumi's user avatar