4
votes
0answers
66 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 ...
7
votes
2answers
290 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 ...
2
votes
0answers
79 views

order of convergence of the conditional entropy (3)

I'm sorry for having open two questions which have been solved by elementary counter-examples provided by @AnthonyQuas. Actually I'm not an expert in information theory and I expected that a positive ...
1
vote
1answer
89 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
votes
0answers
129 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 ...
2
votes
0answers
131 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 ...
5
votes
0answers
169 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 ...
8
votes
1answer
234 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 $P(A_1)=P(A_2)=0$, $f(A)$ ...
2
votes
3answers
213 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 ...
4
votes
1answer
177 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 ...
8
votes
2answers
291 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 ...
4
votes
1answer
232 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 ...
5
votes
0answers
167 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
votes
0answers
203 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 ...
1
vote
0answers
154 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 ...
11
votes
3answers
611 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 ...
6
votes
2answers
701 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 ...
8
votes
3answers
462 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 ...
1
vote
0answers
280 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 ...
4
votes
1answer
162 views

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 ...
12
votes
5answers
2k 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 ...
3
votes
1answer
416 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 ...
10
votes
2answers
721 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 ...
3
votes
1answer
290 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 ...
2
votes
1answer
660 views

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 ...
43
votes
0answers
2k views

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 ...
9
votes
3answers
992 views

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 ...
5
votes
0answers
354 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 ...
2
votes
1answer
267 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). ...
12
votes
2answers
1k views

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 ...
4
votes
1answer
454 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 ...
20
votes
5answers
794 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 ...
4
votes
1answer
324 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 ...
3
votes
1answer
260 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 ...