All Questions
Tagged with ergodic-theory pr.probability
165 questions
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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 ...
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Measure of the rate of convergence for filtration and conditional expectations
This question is cross-posted at MSE with a soon to expire bounty that hasn't generated much discussion.
Let $(\Omega, \mathcal{F},P)$ be a probability space and $(\mathcal{F}_n)_n$ a filtration that ...
- 8,071
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3
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When is the minimal Martin boundary closed?
Let $\Gamma$ be a finitely generated group and $\mu$ a symmetric measure of finite support on $\Gamma$. Let $\partial_{M}\Gamma$ be the Martin boundary of $(\Gamma,\mu)$ and let $\partial^{min}_{M}\...
- 11.6k
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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, ...
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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\}^{\...
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Iterated Circumcircle
Take three noncollinear points (a,b,c), compute the center of their circumcircle x, and replace a random one of a,b,c with x. Repeat. It seems this process may converge to a point, assuming no ...
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Event of positive probability occurs infinitely often in stationary ergodic sequence
Setup:
Suppose $X = \{X_n\}_{n\in\mathbb{Z}}$ is a stationary ergodic proces on the real line and let $A = \prod_{n\in\mathbb{Z}}A_n$ be a Borel measurable set such that
$$
P(X \in A) = P\left(X_n\in ...
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4
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610
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How to generalize normal number theorem
The Borel number theorem states that with respect to Lebesgue measure, almost all real numbers are normal numbers. It is sometimes stated in the context of the compact interval $[0,1]$, where one ...
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Ergodicity of the product Markov chain
$\def\P{\mathsf{P}}$
Let $(X_n)_{n\in\mathbb{Z}_+}$ be a Markov chain with a transition kernel $P(x,dy)$. Consider now a product Markov chain $(X^1_n,X^2_n)_{n\in\mathbb{Z}_+}$ with the transition ...
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Measurable isomorphism between two non-totally ergodic systems
Suppose $(X,\mathcal A,\mu,T)$ is a finite measure-preserving system. Then we define a new measure system $(X^{(K)},\mathcal A^{(K)},\mu^{(K)},T^{(K)})$ defined by $X^{(K)}=X\times \{1,2,...,K\}$ for ...
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On the spectrum of stationary Gaussian process
What is the condition for ergodicity, weakly mixing, and strongly mixing properties of Gaussian process in terms of its spectrum?
In a similar way let us consider a stationary vector valued Gaussian ...
- 421
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Uniqueness of invariant measure for equivalent transition probabilities
Suppose $P(x,dy)$ and $Q(x,dy)$ are two Markov transition kernels on a topological space $E$ equipped with Borel $\sigma$-algebra $\mathcal B(E)$. Suppose for every $x \in E$, $P(x,\cdot)$ and $Q(x, \...
- 17k
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207
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markov processes and ergodic theory
For an ergodic Markov Chain
$$
\frac{1}{N}\sum_{i=1}^n f(X_i) \rightarrow E_\pi[f]
$$
where $\pi$ is the invariant distribution. I am also dealing with a Markovian process (a state space model to ...
- 169
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Weighted sum of i.i.d. random variables
Suppose you have a positive sequence $X_1,X_2,\dots$ of i.i.d. random variables with the property that
$$
\mathbb{E}[\log(X_1)]<\infty.
$$
Is it true that
$$
\limsup_{n\to\infty} e^{-n}\sum_{k=1}^...
- 54.2k
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Can ergodic theory help to prove ergodicity of general Markov chain?
I am a beginner in ergodic theory. I have read some lecture notes(such as this and this) about it in hope that I could find something which helps to prove the ergodicity of some Markov chain taking ...
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Question about martin boundaries of random walks induced on transient subgroups
Suppose $\Gamma$ is a discrete, finitely generated, non-amenable group, and
consider a random walk given by a measure $\mu$.
Assume the measure is symmetric, finitely generated, and the support of
$\...
- 2,964
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2
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492
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Can I use Birkhoff's Ergodic Theorem for Vector Valued Process?
I have a stationary process $\{u_n\}$ and I have a function $f:\mathbb{R}^L\to \mathbb{R}^+$. I want to evaluate the following limit $$\lim_{n\to \infty}\frac{1}{n}\sum_{k=1}^n g(f(\mathbf{u}_{k}))$$ ...
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299
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A weighted ergodic average
According to my simulations, it looks like the number of times that the $N$ first iterates $u_0$, $\ldots$, $u_{N-1}$ of the sequence $(u_n)$ defined here meets an interval $I$ is close to $N|I|$ ...
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139
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weak-* versus entropy growth
General question. Let $\eta_{n}$ be a sequence of invariant measures on $\{0,1,2,...,p-1\}^{\mathbb{N}}$ and $B$ the Bernoulli uniform measure. Knowing that $\eta_{n} \rightarrow B$ in the weak-* ...
- 105k
2
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1
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241
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Shift Invariance of Backward Martingales for tail trivial probability measures
Consider the infinite cartesian product $\Omega=\{0,1\}^{\mathbb{N}}$
as a measurable space endowed with the $\sigma$-algebra $\mathscr{F}$ generated by the cylinder sets and $\sigma:\Omega\to\Omega$ ...
- 29.8k
7
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2
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409
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Estimating entropy conditional to an event
Take for example the measure $\mu(n)=n^2$ on $\{1, \ldots, N\}$ and a random variable $X$ distributed according to the probability obtained by normalizing $\mu$.
Does there exists a constant $K>0$...
- 2,560
4
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106
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Two-side deviations for ergodic sums
Let $(X,\mu)$ be a probability space and $f\colon (X,\mu)\to (X,\mu)$ be an ergodic automorphism. Let $\phi\in L^\infty(X,\mu)$ be such that $\int\phi d\mu=0$.
Suppose that for $\mu$-a.e. $x\in X$, ...
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1
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446
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Importance of Ornstein's isomorphism theorem
"Perhaps the most important parts of the Ornstein theory are criteria for determining whether or not a shift or flow is Bernoulli (a Bernoulli shift, $B_{ct}$ , or $B_{t}^{\infty}$) because it allows ...
- 1,590
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81
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What statistical data/quantities are known about the time spent by a generic orbit of an ergodic system in a fixed set?
By the ergodic theorem, we know that for almost every point, the average time spent by an orbit in a set is equal to the relative measure of that set.
What other information about that time can we ...
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77
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Entropy of the Scenery factor in the $T,T^{-1}$ transformation (RWRS)
The $T,T^{-1}$ transformation is an example of a $K$ automorphism which is not Bernoulli (not isomorphic to a shift of an I.I.D. sequence).
Hoffman in http://www.math.washington.edu/~hoffman/...
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715
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What time does it take for irrational rotations to hit an interval?
Hi,
Consider $\theta_n = (\theta_0 + n \theta) \mod 1$, $\theta$ being an irrational number, and $\theta_0$ an uniform random variable in $(0,1)$. Is there any estimates for the time it will take ...
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321
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Random suborbits of a rotation
Let $u_n = x + n\alpha \pmod 1$ with $\alpha$ irrational. We know that $(u_n)_{n \geq 0}$ is dense in $\mathbb{R}/\mathbb{Z}$ (equivalently $(u_n)_{n \geq 0}$ visits every open interval infinitely ...
- 2,135
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240
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Paths in Pascal's triangle; or balanced $0-1$ initial segments
Here is a problem arising (via a tortuous path) from trying to determine the spectrum of Vershik's adic map on Pascal's triangle (a moderately well-known question: is the spectrum trivial, that is, is ...
- 4,679
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191
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Asymptotically full stationary process
Let $(X_n)_{n \in \mathbb{Z}}$ be a stationary process on a finite set $A$. Say that it is asymptotically full if for every increasing sequence of subsets $B_n \subset A^n$ such that $\dfrac{\#B_n}{\#...
- 17k
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2
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892
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"Typical" convergence rate for the von Neumann mean ergodic theorem
The von-Neumann theorem states that for a unitary operator $U: {\cal H} \mapsto {\cal H}$,
where ${\cal H}$ is a Hilbert space, the following holds:
$$
\lim_{N\to \infty} \frac{1}{N} \sum_{n=1}^N U^n ...
- 445
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2k
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Birkhoff Ergodic Theorem and Ergodic Decomposition Theorem for Continuous-Time Markov Processes
I have a couple of questions regarding ergodicity for Markov processes in continuous time. (In particular, the first question seems like it should be particularly basic, and yet I haven't managed to ...
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209
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On the decay of correlations of an ergodic sequence over the set $X_{0}=0$
The following question arose while I was trying to explore possible further extensions of a CLT by Liverani which I mentioned here already (see this link, I can tell you more details upon request). It ...
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3
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1k
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Central Limit Theorem(s) for irrational rotation
Let $\alpha$ be irrational and $T: S^1 \rightarrow S^1$ be the rotation by $\alpha$. I'm interested in what type of Central Limit Theorem (if any) can hold for sums $Y_n = \frac{1}{\sqrt{n}}\sum_{k=1}^...
- 6,206
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192
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A question related to metric Diophantine approximation
In metric Diophantine approximation you are often interested in finding conditions on $(\phi(q))_{q \geq 1}$ which guarantee that
$$
\left| \alpha - \frac{p}{q} \right| < \frac{\phi(q)}{q}
$$
has ...
- 2,207
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0
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303
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Cesaro mean of products of converging matrices
Let $S$ be a finite set of states. Let $(M_n)$ be a sequence of transitions on $S$; that is, for every natural number $n$, $M_n$ is a non-negative $|S| \times |S|$ matrix whose rows sum up to 1. ...
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594
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limiting distribution of the random walk from irrational rotation
Motivation:
If I recall correctly, the simple symmetric random walk from i.i.d binary steps converges in distribution to the Wiener measure (if scaled with $a_n = \sqrt{n}$). What I am wondering is ...
- 7,514
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trivial map on $\sigma-$algebra $\mod{}0$ is trivial
Hi everyone!
I am currently studying the basic theory of measurable actions and need the following result, which I am not able to prove myself. It is stated without a proof, so probably it should not ...
- 41.1k
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Do regular conditional distributions almost surely assign trivial measure to all members of the conditioning $\sigma$-algebra?
Let $(X,\Sigma)$ be a standard measurable space, let $\rho$ be a probability measure on $(X,\Sigma)$, and let $\mathcal{E}$ be a sub-$\sigma$-algebra of $\Sigma$. We will say that a stochastic kernel $...
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Strange definition of ergodicity
I've already asked this question on math.stack a few days ago and haven't received an answer, so I'm asking here.
In an engineering course, a stationary process was defined to be ergodic if for all $...
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157
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Pointwise convergence of ergodic averages of unconventional conditional expectations
Let $(X_i,Y_i)_{i\in\mathbb{Z}}$ be a finite-valued stationary process whose $\sigma$-algebra of tail events is trivial. Let $\mathcal{F}_n^m$ be the $\sigma$-algebra generated by $X_n,\dots,X_m$ ($n,...
- 191
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348
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"strongly mixing" action on dimers?
In Local Statistics of Lattice Dimers we study a nice familiar object, domino tilings in the plane extending out to infinity.
His paper is going to discuss the frequency of various "motifs" in ...
CommunityBot
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415
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$\{\phi:\int \phi d\mu=0\}$ for a fixed shift invariant $\mu$
Given a shift invariant probability measure $\mu$ on a mixing subshift of finite type.
What are the Lipschitz functions with zero integral with respect to the measure $\mu?$
Clearly any $\phi\in\{-u+...
- 1,805
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698
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Convergence rate of the convolution of almost uniform measures on $\mathbb{Z}_p$
Statement Given a finite abelian group $G$ and two independent random variables $X,Y$ taking values in $G$ and satisfying $d_{TV}(X,U_G)\leqslant \delta$ and $d_{TV}(Y,U_G)\leqslant \delta$ (where $...
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95
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Best convergence rate for convolutions on $\mathbb{Z}_p$
Suppose, that we have sequence of i.i.d variables $X_1,\ldots,X_n$ taking values in $\mathbb{Z}_p$ such that $d_{TV}(X_1,U) < \delta$.
How fast, in terms of $\delta$ and $n$ does the sum $X_1+\...
- 696
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135
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order of convergence of the conditional entropy (2)
Let $X_n$ be a random variable distributed on $A_n:=\{1, \ldots, n\}$ and $g_n\colon A_n \to A_n$ such that $\Pr\big(X_n \neq g_n(X_n)\big) \to 0$. Putting $Y_n=g_n(X_n)$, then by Fano's inequality $$\...
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405
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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 ...
- 3,958
4
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0
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282
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Markov operators and existence of ergodic measures
My question refers to the yesterday's question (see here)
of John Learner and goes as follows:
Can we deduce the existence of an ergodic measure if we know that an invariant measure exists, but the ...
CommunityBot
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221
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Quasicompactness of transfer operators associated to IID matrix products
Let $P^1$ denote one-dimensional real projective space, and for each $A \in GL(2,\mathbb{R})$ let $\overline{A}$ denote the homeomorphism of $P^1$ induced by $A$. I am currently reading a paper which ...
- 6,206
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357
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Random variables invariant under almost automorphisms.
Let $\Omega$ be a standard atomless probability space, we can assume $\Omega=(0,1)$ with Lebesgue measure. A bijection $f:\Omega/A_1\to\Omega/A_2$ is almost automorphism, if $P(A_1)=P(A_2)=0$, $f(A)$ ...
- 1,354
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3
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702
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The property of a Markov measure
Given $\sigma$ a shift map, $m$ - a Markov measure, $C_a$, $C_b$ - cylinder sets.
Suppose $P \in C_b$. The problem is to show the following
\begin{equation}
m(C_a \cap \sigma^{-1}(P)) = \frac{m(C_a \...
- 2,560