# Questions tagged [mixing]

For questions about mixing in ergodic or probability theory.

119 questions
Filter by
Sorted by
Tagged with
109 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 ...
67 views

104 views

74 views

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 ...
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 ...
38 views

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 ...
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: ...
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 ...
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 $Y_i = \sum_{k=1}^i V_i$ ...
Just to give a context, define the mixing coefficient by $$\alpha(j)=\sup_T\sup_{1\leq k\leq T-j}\sup\left\{\lvert P(A\cap B)-P(A)P(B)\rvert: B\in\mathcal{F}_{T,1}^{k},A\in\mathcal{F}_{T,k+j}^T\right\... 0answers 36 views ### Understanding ergodic/mixing group actions and flows Setup: Let X be some probability space with probability measure \mu (so \mu(X)=1). And let G be a one real perameter topological group so G=\{g_s:s\in\mathbb{R}\}, and for the sake of ... 1answer 177 views ### Check the \alpha-mixing conditions: Stochastic process I want to determine if \{Y_{t,T},X_{t,T}\}, t=1,\dotsc,T; T\geq 1, is \alpha-mixing for some special cases. Let X_{t,T}\sim i.i.d. U[0,1] and Y_{t,T}=1 (degenerate random variable); Y_{t,T}=... 0answers 46 views ### \phi-mixing rate and AR process The general definition of the \phi-mixing coefficient is as follows: \begin{equation} \phi(\sigma_{1},\sigma_{2}) = \underset{B\in\sigma_{1},A\in\sigma_{2}}{\sup} P(A|B)-P(A) \end{equation} Then ... 1answer 42 views ### Estimating the interruption rate of a mixed poisson process. For a regular Poisson process, we know that the inter-arrival distribution, T is exponential with rate \lambda and the number of events in any interval, N(t) is Poisson with mean \lambda t and ... 1answer 41 views ### Upper bound for variance of kernel averages I'm reading Hansen's (2008, p. 729) Theorem 1 where he bounds the variance of averages of the form$$\hat\Psi(x)=\frac1{Th}\sum\limits_{t=1}^T Y_t K\bigg(\frac{x-X_t}h\bigg)$$given that \{(Y_t,X_t)\... 0answers 41 views ### Point process theory: Proof that strong mixing implies mixing. Here's the problem: I'm working on a paper that says that strong mixing condition for stationary point processes implies the process to be mixing, but it never proves it (the paper is Ivanoff, Central ... 1answer 122 views ### If \alpha irrational, then F(x,y)=(x+\alpha,x+y)\mod1, T^{2}\to T^{2} preserves Lebesgue measure and is not weak mixing Let \alpha be an irrational number. How do I prove that$$F(x,y)=(x+\alpha,x+y)\mod1,\qquad T^{2}\to T^{2}$$has the following properties: Lebesgue measure invariant: \lambda(F^{-1}([a,b)\times[c,... 2answers 142 views ### Decay of tail of a function - relation to integrability Possibly a basic question, but I could not find anything online. Suppose we have a bounded metric space (X,d) with a Borel probability measure \mu. If we have an integrable function f:X\to \... 1answer 65 views ### Is a transformed mixing process is still mixing? Let \{X_t\}_{t=1}^\infty be a stationary \alpha-mixing process, where X_t \in \mathbb{R}^m and let f:\mathbb{R}^m \rightarrow \mathbb{R}^n  be a Borel-measurable function. Is the transformed ... 0answers 57 views ### Alternative definitions of \beta mixing coefficients I am looking at to two different ways to define \beta mixing coefficients and I am struggling to understand how and why they are the same. R.Bradley defines:$$\beta(\mathcal{A, B}) := \sup\frac{...
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 ...