All Questions
24 questions
2
votes
0
answers
83
views
Random time change and ergodicity
I guess it is a standard question in ergodic theory but I failed to find any reference to similar problems and I have no clue on how to tackle it.
Let $(B_{t})_{t\in \mathbb{R}}$ be a standard ...
- 21
3
votes
2
answers
223
views
Measures with superexponential moments on finitely generated groups
Let $\Gamma$ be an infinite finitely generated group and let $\nu$ be a measure on $\Gamma$ which generates a transient random walk. I was reading this paper, and the authors prove many of their ...
2
votes
1
answer
266
views
Ergodicity of linear dynamical systems and convergence of covariance matrices
Let $z(n+1)=Bz(n)+\xi(n+1)$ be an $N$-dimensional linear dynamical system with $\left(\xi(n)\right)_{n\in\mathbb{N}}$ being i.i.d. with $\xi(n)\sim\mathcal{N}(0,\Sigma_{\xi})$.
Assumptions: a) The ...
- 767
20
votes
1
answer
2k
views
Roadmap to Ergodic Theory
I have recently been interested in going deeper into ergodic theory, beyond an introductory level of knowledge. Background wise, my training has mostly been in stochastic analysis, and I have a ...
- 6,195
2
votes
0
answers
313
views
Correlation decay rate
Let $T$ be a continuous transformation of a probability measure space $(X,\mathcal{B}(X),\mu)$ and
$\varphi ,\phi \in L^2(\mu)$ (so-called observable) . The correlation function of $\varphi ,\phi$ (a ...
- 136
4
votes
1
answer
363
views
Maximal ergodic inequality
A map $f: X \to X$ preserves an ergodic probability $\mu$, i.e., $\mu \circ f^{-1}=\mu$ and for any $\phi: X \to \mathbb{R}$ with $\int \phi d\mu=0$,
$$\frac{1}{n} \sum_{i \le n} \phi \circ f^i \to 0 \...
- 43
1
vote
1
answer
137
views
Ergodic theorem on limit of periodic transformations?
Suppose $(X,\mu)$ is a probability space, and $T_n, n \in \mathbb N$, is a sequence of periodic measure preserving transformations. For $x \in X$ and $f : X \to \mathbb R$, let $\mathrm{avg}_{f,n}(x)$...
- 18.6k
3
votes
0
answers
153
views
Metropolis-Hastings sampling as a group action
Suppose that you have a topological space $\Omega \subset \mathbb R^n$ accompanied a measure $\mu$ and you're running an iterative sampling algorithm like Metropolis-Hastings. To sample you choose a ...
4
votes
0
answers
95
views
When the Jacobian of unstable measure converges
Let $T:X \to X$ be a hyperbolic map on the compact metric space $X$. Hyperbolicity means that $T$ has local stable and unstable sets with uniform exponential bounds, which satisfy a local product ...
- 1,043
5
votes
0
answers
183
views
Central limit theorem versus entropy in dynamical systems context
A dynamical system $(S^1,T, \mu)$, $T_* \mu=\mu$, $T$ ergodic, $S^1$ is circle. Assume it has central limit theorem.
Want to know the relation between its measure-theoretic entropy $h_{\mu}(T)$ and ...
- 553
3
votes
0
answers
188
views
Invariant subspaces of Markov operators
I am currently working on some kind of graph theoretic problem and the following question came up:
Suppose you have a Markov operator $T$ on $\ell^\infty$, that is a positive, bounded operator such ...
- 381
2
votes
0
answers
113
views
Characterizing the relationship between element-wise Markov transitions and the full-conditionals of the stationary distribution
Consider a $p$ dimensional random variable with a discrete support. Consider a Markov transition kernel on the state space that is defined in terms of element-wise transition distributions.
One can ...
- 2,791
4
votes
2
answers
201
views
Uniform convergence of averages for stationary ergodic process
Let $\{X_t, t\in\mathbb R\}$ be a well-behaved$^*$ stationary ergodic process.
I'm interested in the uniform convergence of averages:
$$
\sup_{|x|\le R_n} \left|\frac1{2n}\int_{x-n}^{x+n} X_t dt - \...
- 1,533
2
votes
2
answers
242
views
iid random operator and its spectrum
consider an insteresting question:
given Banach Space $ \mathcal{B}$, independent identical distribution random operator on $ \mathcal{B}$: $ (T_i)_{i \ge 1} $, where operator space is endowed with ...
- 553
3
votes
0
answers
95
views
Empirically random, quickly multiplicable matrices
I have encountered a need for fast computation of a transformation $Ax$ where $A\in \mathbb{C}^{K\times N},\ K\sim 10^7,\ N\sim 10^3$ is designed, and $x\in \mathbb{C}^N$ has iid $\mathcal{CN}(0,1)$ ...
1
vote
1
answer
404
views
Does Irreducibility holds for the Ergodic non-stationary Markov chain?
In the stationary case, I know that if the chain is irreducible and aperiodic, it is Ergodic. But in the non-stationary case, i can not comprehend the content deeply. I want to know if Irreducibility ...
- 121
5
votes
0
answers
143
views
Law of Large Numbers for the Tasep from a Bernoulli Configuration (Rost's Theorem)
Let $(\eta_{t}^{\rho})_{t\geq 0}$ be a totally asymmetric simple exclusion process (TASEP) from an initial configuration distributed according to the Bernoulli measure $\nu_{\rho}$ on $\{0,1\}^{\...
1
vote
1
answer
193
views
Optimal joint coupling of all probability measures on a 3 point space
I am looking for any remotely related reference for the following problem, for which I have not the least clue what techniques would be useful.
Consider a discrete probability space $\Omega = \{x, y, ...
- 4,466
4
votes
1
answer
340
views
On the spectrum of stationary Gaussian process
What is the condition for ergodicity, weakly mixing, and strongly mixing properties of Gaussian process in terms of its spectrum?
In a similar way let us consider a stationary vector valued Gaussian ...
- 421
10
votes
2
answers
2k
views
Birkhoff Ergodic Theorem and Ergodic Decomposition Theorem for Continuous-Time Markov Processes
I have a couple of questions regarding ergodicity for Markov processes in continuous time. (In particular, the first question seems like it should be particularly basic, and yet I haven't managed to ...
- 708
21
votes
3
answers
1k
views
Central Limit Theorem(s) for irrational rotation
Let $\alpha$ be irrational and $T: S^1 \rightarrow S^1$ be the rotation by $\alpha$. I'm interested in what type of Central Limit Theorem (if any) can hold for sums $Y_n = \frac{1}{\sqrt{n}}\sum_{k=1}^...
- 3,323
4
votes
0
answers
405
views
Reference request: stationary measures as convex combinations of ergodic measures
Does anyone know a good reference for the fact that a stationary probability measure is a convex combination of the stationary and ergodic probability measures?
I have found some references for the ...
- 710
8
votes
3
answers
749
views
non-integrable subadditive ergodic theorem
Dear MO_World,
I have (another) question about relaxing the assumptions in the sub-additive ergodic theorem. Apologies if this is something I should know already...
There are a number of statements ...
- 23.2k
25
votes
6
answers
6k
views
Proof of Krylov-Bogoliubov theorem
Where can I find a proof (in English) of the Krylov-Bogoliubov theorem, which states if $X$ is a compact metric space and $T\colon X \to X$ is continuous, then there is a $T$-invariant Borel ...
- 897