Questions tagged [mixing]

For questions about mixing in ergodic or probability theory.

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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 ...
3
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1answer
67 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 ...
1
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0answers
22 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). ...
1
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1answer
38 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 ...
3
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1answer
169 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\...
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1answer
25 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 ...
2
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0answers
22 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 ...
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0answers
23 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 ...
3
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1answer
43 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 $T$ ...
2
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0answers
26 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. ...
2
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0answers
60 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 ...
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1answer
29 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. ...
2
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0answers
53 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 ...
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0answers
21 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 \...
2
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1answer
47 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 ...
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0answers
21 views

What part of the Lyapunov spectrum is conserved in a time-delay embedding under Takens' theorem?

My understanding is that time-delay embeddings are often used to estimate the maximum Lyapunov exponent of a chaotic system for which we may not have full state measurements. It seems that you cannot ...
1
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1answer
29 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 ...
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1answer
49 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+...
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1answer
104 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\...
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0answers
93 views

Mixing property of stationary Process, only finite dimensional marginal distributions needed??

Let $X=(X_t)_{t\in\mathbb{R}}$ be a stationary process, i.e. the shifted process $X^s=(X_t^s)_{t\in\mathbb{R}}=(X_{t+s})_{t\in\mathbb{R}}$ has the same distribution as $X=X^0$, and assume that $X$ has ...
4
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0answers
170 views

Can a Markov chain converge faster than a geometric rate?

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 \...
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0answers
52 views

Lower bound on mixing time to lazy random walk on complete graphs

Let $K_n$ be complete graph with $n$ vertices and $$d^{\infty }(t) := \max_{x,y\, \in V} \left|\frac{P^{t}(x,y)}{\pi(y)}-1\right|$$ and $$t^{\infty}_{\text{mix}} := \min \{t \geq 0 : d^{\infty }(t) \...
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1answer
74 views

Markov chain mixing in finite time

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 \...
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1answer
55 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-...
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0answers
38 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 ...
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1answer
34 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) ...
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1answer
86 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
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1answer
174 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 ...
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0answers
37 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 $...
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0answers
112 views

Mixing time of a biased random walk on a line segment

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 ...
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0answers
43 views

Counterexample to Davydov inequality for strictly stationary mixing sequence

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 ...
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0answers
38 views

Finiteness of expectation of product

Suppose $(X_1, X_2), (Y_1, Y_2)$ are two random vectors in $\mathbb{R}^2$ with the same (joint) distributions and suppose $\mathbb{E}[X_1|X_2]=0$ and $\mathbb{E}[X_1^2|X_2]=1$ and $\mathbb{E} X_2^4<...
3
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1answer
74 views

A question about the conjugation

I'm studying something about the Conjugation and my reference propose to prove that the property of being topologically mixing is preserved by conjugation. Obviously the definition of topologically ...
1
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1answer
275 views

Davydov's inequality and strongly mixing sequence.

Let $\{X_i\}$ be an $\alpha$-mixing random process with coefficients $\alpha(k)$ satisfying $\alpha(k)\leq Ca^k$ for some positive constants $a<1$ and $C$. Given that for any $i\in \mathbb{N}$, $E\...
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0answers
67 views

Relation between ergodicity and weakly mixing

A transformation $T$ on $X$ is ergodic iff for any two measurable sets $U$ and $V$ holds: $\lim_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1} m(T^{-j}U\cap V)=m(U)m(V)$, (or equivalently iff every ...
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0answers
33 views

characterization of lightly mixing

In Friedman and King's book "Rank one lightly mixing" and in Wenyu Chen's paper "the notion of mixing and rank one examples", the authors recall the definition of a lightly mixing transformation T: ...
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0answers
36 views

Relaxation time and poincare inequality

This question relates to Levin, Peres, and Wilmer’s book, and specifically to http://www.statslab.cam.ac.uk/~beresty/Articles/mixing2.pdf. I am looking for a proof to the following claim: I do not ...
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1answer
126 views

How to compute the strong mixing coefficient of a sequence

I am having problems with computing alpha-mixing rate in a sequence of random variables as the notation and definition seems a bit abstract to me. For example, suppose that $Y_i = \sum_{k=1}^i V_i$ ...
3
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1answer
34 views

Sigma-algebra inclusion and mixing processes

Just to give a context, define the mixing coefficient by $$\alpha(j)=\sup_T\sup_{1\leq k\leq T-j}\sup\left\{\lvert P(A\cap B)-P(A)P(B)\rvert: B\in\mathcal{F}_{T,1}^{k},A\in\mathcal{F}_{T,k+j}^T\right\...
3
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0answers
36 views

Understanding ergodic/mixing group actions and flows

Setup: Let $X$ be some probability space with probability measure $\mu$ (so $\mu(X)=1$). And let $G$ be a one real perameter topological group so $G=\{g_s:s\in\mathbb{R}\}$, and for the sake of ...
3
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1answer
177 views

Check the $\alpha$-mixing conditions: Stochastic process

I want to determine if $\{Y_{t,T},X_{t,T}\}, t=1,\dotsc,T; T\geq 1$, is $\alpha$-mixing for some special cases. Let $X_{t,T}\sim i.i.d. U[0,1]$ and $Y_{t,T}=1$ (degenerate random variable); $Y_{t,T}=...
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0answers
46 views

$\phi$-mixing rate and AR process

The general definition of the $\phi$-mixing coefficient is as follows: \begin{equation} \phi(\sigma_{1},\sigma_{2}) = \underset{B\in\sigma_{1},A\in\sigma_{2}}{\sup} P(A|B)-P(A) \end{equation} Then ...
2
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1answer
42 views

Estimating the interruption rate of a mixed poisson process.

For a regular Poisson process, we know that the inter-arrival distribution, $T$ is exponential with rate $\lambda$ and the number of events in any interval, $N(t)$ is Poisson with mean $\lambda t$ and ...
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1answer
41 views

Upper bound for variance of kernel averages

I'm reading Hansen's (2008, p. 729) Theorem 1 where he bounds the variance of averages of the form $$\hat\Psi(x)=\frac1{Th}\sum\limits_{t=1}^T Y_t K\bigg(\frac{x-X_t}h\bigg)$$ given that $\{(Y_t,X_t)\...
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0answers
41 views

Point process theory: Proof that strong mixing implies mixing.

Here's the problem: I'm working on a paper that says that strong mixing condition for stationary point processes implies the process to be mixing, but it never proves it (the paper is Ivanoff, Central ...
1
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1answer
122 views

If $\alpha$ irrational, then $F(x,y)=(x+\alpha,x+y)\mod1$, $T^{2}\to T^{2}$ preserves Lebesgue measure and is not weak mixing

Let $\alpha$ be an irrational number. How do I prove that $$F(x,y)=(x+\alpha,x+y)\mod1,\qquad T^{2}\to T^{2}$$ has the following properties: Lebesgue measure invariant: $\lambda(F^{-1}([a,b)\times[c,...
2
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2answers
142 views

Decay of tail of a function - relation to integrability

Possibly a basic question, but I could not find anything online. Suppose we have a bounded metric space $(X,d)$ with a Borel probability measure $\mu$. If we have an integrable function $f:X\to \...
2
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1answer
65 views

Is a transformed mixing process is still mixing?

Let $\{X_t\}_{t=1}^\infty$ be a stationary $\alpha$-mixing process, where $X_t \in \mathbb{R}^m$ and let $f:\mathbb{R}^m \rightarrow \mathbb{R}^n $ be a Borel-measurable function. Is the transformed ...
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0answers
57 views

Alternative definitions of $\beta$ mixing coefficients

I am looking at to two different ways to define $\beta$ mixing coefficients and I am struggling to understand how and why they are the same. R.Bradley defines: $$\beta(\mathcal{A, B}) := \sup\frac{...
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1answer
33 views

Eigenbasis and Linear Combination for Mixing Times

I am working on something for Markov Chain mixing times and I am stuck on a piece of linear algebra. Assume an $n \times n$ matrix M has all simple eigenvalues. Therefore, there is a basis of ...