All Questions
Tagged with ergodic-theory pr.probability
165 questions
5
votes
1
answer
225
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Subadditive Kingmans theorem for lattices.
I am looking for a multidimensional version of Kingman's subadditive theorem. I found this but it is not exactely what I need.
I would rather have something like that:
Let us consider $\mathbb{Z}^...
- 1,491
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
6
votes
1
answer
805
views
Ergodicity of Convoluted White Noise
I have a question regarding ergodicity in infinite dimensional spaces.
Let $\mathcal{D}$ be the space of distributions on a Schwartz space, and let $\mu$ be the white noise process which exists by ...
- 617
13
votes
2
answers
1k
views
Non-integrable ergodic theory
Can anyone help me out with proofs/counterexamples? I'm working on an operator-valued multiplicative ergodic theorem and need what may(?) be a well-known fact. This fact (if true) would help me get ...
- 23.2k
2
votes
2
answers
557
views
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 ...
- 23
2
votes
1
answer
1k
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Given a probability \mu, can we always find a transformation T s.t. \mu is T-invariant?
It is true that, under some conditions, given a measure-preserving transformation $T$, we can always construct a $T$-invariant probability. I am wondering whether we can do a converse. See Parry's ...
51
votes
0
answers
2k
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Alternating colors on a line: infinitely often or converge?
Suppose we have intervals of alternating color on $\mathbb{R}$ (say, red / blue / red / blue / …). All intervals have independent length, with all red intervals distributed as $\mathbb{P}_{R}$, all ...
- 1,010
16
votes
3
answers
2k
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A random walk on random lines
I am wondering if this random walk remains finite with positive probability.
Start with three lines $A,B,C$ that are extensions of an equilateral triangle.
Let $p_0$ be one corner. Generate a line $...
- 151k
7
votes
0
answers
717
views
Is there a continuous-time version of Kingman's subadditive decomposition theorem?
Kingman's subadditive ergodic theorem (see this article) states that if $x_{m,n}$ is a real valued process indexed on the set of pairs of non-negative integers $m < n$ satisfying:
$x_{l,n} \le x_{...
- 4,304
2
votes
1
answer
349
views
exactness of the Gauss transformation
Dear all,
I would like to know if the Gauss transformation T(x) = fractional part of 1/x, x in (0,1) (with the Gauss invariant probability measure) is an exact endomorphism (in the sense of Rokhlin). ...
17
votes
2
answers
2k
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Random walk is to diffusion as self-avoiding random walk is to ...?
One can view a random walk as a discrete process whose continuous
analog is diffusion.
For example, discretizing the heat diffusion equation
(in both time and space) leads to random walks.
Is there a ...
- 151k
6
votes
1
answer
819
views
Birkhoff ergodic theorem for dynamical systems driven by a Wiener process
At the risk of asking a stupid question I have the following problem.
Suppose I have a measure preserving dynamical system $(X, \mathcal{F}, \mu, T_s)$, where
$X$ is a set
$\mathcal{F}$ is a sigma-...
- 617
20
votes
5
answers
1k
views
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 ...
- 151k
5
votes
1
answer
400
views
Is the average first return time of a partitioned ergodic transformation just the number of elements in the partition?
For some reason my thinking is very fuzzy today, so I apologize for the following rather silly question below...
Let $T$ be an ergodic transformation of $(X,\Omega, \mathbb{P})$ and let $X$ be ...
- 15.4k
3
votes
1
answer
295
views
Finitarily Markovian Finite Factors of Bernoulli Schemes
By processes, I mean discrete, stationary stochastic processes, that is $(X,\mathcal{U},\mu,T)$ where $X$ is the set of doubly infinite sequences of some alphabet $A$, $\mathcal{U}$ is the $\sigma$-...
- 325