All Questions
Tagged with pr.probability ergodic-theory
165 questions
1
vote
1
answer
222
views
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, \...
- 985
2
votes
0
answers
207
views
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
1
vote
1
answer
466
views
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}^...
- 491
2
votes
4
answers
610
views
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 ...
- 121
3
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0
answers
157
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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
6
votes
2
answers
3k
views
What's the probability distribution of a deterministic signal or how to marginalize dynamical systems? (functional integrals in probability theory)
Because I still have no idea how it is possible for me to write down seemingly important equations ... that don't make any sense (at least for me) and because I haven't got any helpful comment so far, ...
- 1,026
2
votes
0
answers
299
views
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|$ ...
- 2,560
1
vote
0
answers
139
views
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-* ...
2
votes
1
answer
241
views
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$ ...
- 2,044
9
votes
1
answer
950
views
Sort-of converse of Kolmogorov zero-one theorem
Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space. The Kolmogorov zero-one theorem states that
Suppose we have independent random variables $X_1, X_2, ...$. Then $\forall \ A \in \bigcap_n ...
- 247
4
votes
1
answer
106
views
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$, ...
- 1,060
5
votes
0
answers
81
views
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 ...
user76098
7
votes
2
answers
409
views
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
5
votes
1
answer
446
views
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 ...
- 395
2
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0
answers
77
views
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/...
- 326
7
votes
2
answers
321
views
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 ...
- 2,560
2
votes
2
answers
492
views
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}))$$ ...
5
votes
0
answers
240
views
Paths in Pascal's triangle; or balanced $0-1$ initial segments
Here is a problem arising (via a tortuous path) from trying to determine the spectrum of Vershik's adic map on Pascal's triangle (a moderately well-known question: is the spectrum trivial, that is, is ...
- 4,679
0
votes
1
answer
191
views
Asymptotically full stationary process
Let $(X_n)_{n \in \mathbb{Z}}$ be a stationary process on a finite set $A$. Say that it is asymptotically full if for every increasing sequence of subsets $B_n \subset A^n$ such that $\dfrac{\#B_n}{\#...
- 2,560
5
votes
2
answers
892
views
"Typical" convergence rate for the von Neumann mean ergodic theorem
The von-Neumann theorem states that for a unitary operator $U: {\cal H} \mapsto {\cal H}$,
where ${\cal H}$ is a Hilbert space, the following holds:
$$
\lim_{N\to \infty} \frac{1}{N} \sum_{n=1}^N U^n ...
- 445
3
votes
0
answers
209
views
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 ...
- 486
2
votes
0
answers
192
views
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
2
votes
0
answers
303
views
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. ...
- 745
8
votes
2
answers
594
views
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 ...
- 307
11
votes
2
answers
2k
views
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 ...
- 221
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2
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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
8
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3
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834
views
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 $...
- 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
9
votes
1
answer
626
views
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 $...
- 935
3
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0
answers
157
views
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
5
votes
1
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348
views
"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 ...
- 22.8k
1
vote
2
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415
views
$\{\phi:\int \phi d\mu=0\}$ 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
3
votes
0
answers
95
views
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
7
votes
2
answers
698
views
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 $...
- 696
1
vote
1
answer
135
views
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 $$\...
- 2,560
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
4
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0
answers
282
views
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 ...
- 141
5
votes
0
answers
221
views
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
9
votes
1
answer
357
views
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 $P(A_1)=P(A_2)=0$, $f(A)$ ...
- 781
2
votes
3
answers
702
views
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 \...
- 23
4
votes
1
answer
213
views
Practical way to check for geometric convergence
Target distribution is multimodal, 24 dimensions, continuous state space. For MCMC integration (MH sampler) I use a manually tuned proposal distribution.
When I measure the convergence rate ...
- 101
13
votes
2
answers
715
views
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 ...
- 411
5
votes
1
answer
437
views
Stationary, ergodic measures from the structuralist point of view
Stationary, ergodic measures are a class of objects very familiar to probabilists. In a sense, these are the weakest generalization of the classic case of independent, identically distributed random ...
- 8,512
6
votes
0
answers
301
views
Generating stationary, ergodic random fields on a homogeneous space
Consider a homogeneous space $M$, which for the sake of concreteness, let's take to be $M = \mathbb R^d$. Fix some space $A$, and consider the space of functions $X = C(M,A)$, along with its Borel $\...
- 8,512
11
votes
1
answer
1k
views
resampling over Bowen balls
Hello MO World
I'm working on a paper involving embedding your favourite measure-preserving transformation into a topological model (think Krieger generator theorem: embedding in a full shift) and ...
- 23.2k
1
vote
0
answers
179
views
Entropy of Bernoulli walks on semi-groups.
Consider the Fibonacci semi-group $<L,R|LRR=RLL>$ with a Bernoulli walk $P(R)=p, P(L)=1-p$. Is the entropy $H(p)$ an unimodal function with maximum at p=0.5? Is this true for all finitely ...
- 1,648
12
votes
3
answers
891
views
Looking for at least one beautiful and not too technical result in asymptotic group theory
We have a student seminar devoted to the problems of asymptotic group theory with some connections to ergodic theory and measure theory in general. Each talk concerns one of the problems of this ...
11
votes
2
answers
2k
views
De Finetti's theorem, the pointwise ergodic theorem, and reverse martingales
De Finetti's theorem says that an exchangeable sequence of random variables $X_i$ is a mixture of i.i.d. random variables. In other words, if $\mu$ is a measure on $\mathbb{R}^\infty$ that is ...
- 6,287
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
0
votes
0
answers
337
views
What is the mean-value of a particular exponential sum related to the non-trivial zeros of Riemann's zeta function?
This question arose from an earlier one and the MO user's useful answers there: What are the values of the derivative of Riemann's zeta function at the known non-trivial zeros? (which is not a ...
- 2,480