Questions tagged [mixing]

For questions about mixing in ergodic or probability theory.

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Time inhomogeneous Markov Chain

I am currently studying a time inhomogeneous Markov chain with the following behavior: There exists an underlying irreducible, aperiodic and reversible Markov Chain $C$ but at each point in time only ...
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83 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 ...
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146 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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27 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
35 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
27 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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34 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
24 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
53 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 ...
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35 views

weak dependence, strong mixing and martingales

I am not very familiar with concepts from statistics such as weak dependence and strong mixing (I believe it is also called $\alpha$-mixing). Let's stay in discrete time ($\mathbb{N}_0$) and, just to ...
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1answer
90 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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31 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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55 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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29 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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31 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<...
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1answer
66 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 ...
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1answer
68 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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37 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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25 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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29 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
59 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 $Y_i = \sum_{k=1}^i V_i$ ...
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24 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\...
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32 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 ...
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1answer
63 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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18 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 ...
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1answer
32 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
33 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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22 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 ...
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1answer
99 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,...
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2answers
70 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 \...
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1answer
44 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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52 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
30 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 ...
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1answer
41 views

Error in proof of $L_p$ equivalent weak-mixing conditions.

In page 45 of Peter Walters' "Introduction to Ergodic theory" there is a theorem which states that: Theorem: $T:X \to X$ is weak-mixing if and only if for all $f,g \in L_2(X)$ we have that $$ \...
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1answer
95 views

Is linear combination of strong mixing process still strong mixing?

Suppose the process $\left\{X_{t}:t\in Z\right\}$ is absolutely continuously distributed and strong mixing with the coefficient $\alpha_{X}(s) \rightarrow 0$ defined as \begin{equation} \alpha_{X}(s) \...
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1answer
36 views

Is the infinitely distributed lag process strong mixing?

Suppose the process $\{X_{t}:t\in Z\}$ is absolutely continuous distributed and strong mixing with the coefficient $\alpha(s)$ defined as \begin{equation} \alpha(s) \equiv \sup \{ |P(A\cap B) - P(A)P(...
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1answer
26 views

Expressing a CDF with positive support as a mixture of two components

Consider the two-component mixture $$ F(z)=\lambda F_1(z)+(1-\lambda)F_2(z) $$ where all the $F$'s are CDFs and $\lambda\in [0,1]$. A1: Assume that $F(z)=0$ $\forall z\leq 0$. Claim: A1 implies ...
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1answer
88 views

Overview of the different types of mixing sequences

I have come across the terms strong mixing, $\alpha$-mixing, $\beta$-mixing, $\phi$-mixing, $\rho$-mixing. Could somebody please compile an answer that would summarize their definition in some clear ...
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1answer
182 views

Downsizing baby formula [closed]

I couldn't find a stack exchange for babies, so here goes. We mix ten scoops of formula into 150ml of water to get 200ml of milk. What is the combo I need to make 150ml of milk? I tried mixing 7.5 ...
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1answer
83 views

How to get conditional from mixed normals (Bayes rule)?

I need help with something I am getting confused on (this is not a homework problem). If you have a mixture of 2 normal distributions, how do I use Bayes rule with another normal to get a conditional ...
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78 views

Mixing time of personalized PageRank random walk

Suppose I have a weighted graph $G$, and let $M$ be the Markov chain corresponding to a random walk on $G$. Let $t_{\text{mix}, M}(\epsilon)$ be the $\epsilon$-mixing time of $M$. Now let $M'$ be ...
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86 views

Density limits and sets with positive lower density

This question is informed by a lemma that is needed in pursuit of an ergodic theory result specifically to do with weak mixing. We first need two definitions before we can state the proposition in ...
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1answer
86 views

Ergodic system counterexample

Can anyone think of an easy example when $$(X, \mathcal{B}, \mu, T)$$ is an ergodic measure-preserving system but $$(X, \mathcal{B}, \mu, T^k)$$ is not ergodic for $k$ a positive integer.
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60 views

Mixing time of a random walk

I have a very basic doubt about mixing time of the graph. I want to know how the mixing time reduces as I add edge to a connected graph $G$. Is there a possibility of the mixing time of the random ...
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134 views

Example of a weak mixing that is not Topological Mixing

A continuous map $f:X\to X$ is called Topologically transitive or TT if for every pair of non-empty open sets $U,V$ in $X$ there exists $n\in \Bbb{N}$ such that $f^n(U)\cap V \neq \emptyset$. A ...
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68 views

MLE of mixing parameters for “n” single-variable Gaussian distributions

I am looking to derive the maximum likelihood estimator of the mixing parameters for "n" single-variable Gaussian distributions. The complete model density is: $$ f(x) = \sum_{k=1}^m \lambda_k f_k(...
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78 views

Are functions of rho-mixing random variables rho-mixing?

In Bradley (1986, p. 170), it is stated that rho-mixing implies strong mixing. In White's book on asymptotic theory (p. 50, T3.49 and P3.50), it is stated that if $Z_t$, $X_t$ and $\epsilon_t$ are ...
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91 views

Show that if dynamical system $T$ is weakly mixing, then $T^n$ is weakly mixing.

Show that if dynamical system $(X,T)$ is weakly-mixing, then $(X, T^n)$ is weakly mixing. I am using this definition of weakly mixing: we say that system is weakly mixing if system $(X^2, T \times T$)...
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1answer
328 views

What is mixing condition in Markov Decision Processes?

What is a mixing condition of an MDP? I'm reading a paper called Experts in a Markov Decision Process, and it says Before we provide our algorithm a few definitions are in order. For every ...
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1answer
426 views

Intuitive explanation of the spectral gap in context of Markov Chain Monte Carlo (MCMC)

I'm learning about Markov Chains Monte Carlo methods and mixing times, and could use some help understanding the concept of the Spectral Gap and why / how it relates to the mixing times. Thus far ...