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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 ...
Quinn Culver's user avatar
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 ...
12 votes
1 answer
1k views

Riesz–Markov–Kakutani representation theorem for compact non-Hausdorff spaces

Let $X$ be a compact Hausdorff topological space, and $\mathcal C^0 (X) = \{f:X\to\mathbb{R}; \ f \text{ is continuous }\}$. It is well known that for any bounded linear functional $\phi: \mathcal C^...
Matheus Manzatto's user avatar
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 ...
BCLC's user avatar
  • 247
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 $P(A_1)=P(A_2)=0$, $f(A)$ ...
Bogdan's user avatar
  • 781
8 votes
3 answers
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 $...
Julian Newman's user avatar
7 votes
1 answer
253 views

Are all quasi-regular points on Polish spaces generic points?

Let $X$ be a Polish space and $T\colon X\to X$ be a continuous map. We say that a point $x\in X$ is quasi-regular if for every bounded continous function $\varphi\colon X\to\mathbb{R}$ the sequence $...
Dominik Kwietniak's user avatar
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 $\...
Tom LaGatta's user avatar
  • 8,512
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 ...
Tom LaGatta's user avatar
  • 8,512
4 votes
1 answer
227 views

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 ...
Marc's user avatar
  • 479
4 votes
1 answer
352 views

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 ...
aduh's user avatar
  • 869
4 votes
1 answer
446 views

Birkhoff ergodic theorem for ergodic Markov processes

This question was previously posted on MSE. This question might be easy but I am really stuck on it. Let $M$ be compact metric space and $\mathcal B(M)$ the Borel $\sigma$-algebra of M. Consider the ...
Matheus Manzatto's user avatar
3 votes
2 answers
163 views

Questions about some properties of random probabilities and random expectations

Let $(\Omega, \mathcal{A}, \mathbb P)$ be a probability space with $\mathcal{A}$ countably generated, and let $P: \mathcal{A} \times \Omega \to [0,1]$ be a random probability measure. By that I mean $...
aduh's user avatar
  • 869
3 votes
2 answers
250 views

Existence of a positive measurable set with disjoint preimage under iterated transformation

Let $(X,\mathcal B,\mu)$ be a atomless probability measure space and $T:X\to X$ be a non-singular transformation such that $\mu\left({x\in X: T^n(x)=x}\right)=0$ for every $n\ge 1$. Let $A\in \mathcal ...
abcdmath's user avatar
  • 105
3 votes
1 answer
372 views

Attractors in random dynamics

Let $\Delta$ be the interval $[-1,1]$, then we can consider the probability space $(\Delta , \mathcal{B}(\Delta),\nu)$, where $\mathcal{B}(\Delta)$ is the Borel $\sigma$-algebra and $\nu$ is equal ...
Matheus Manzatto's user avatar
3 votes
1 answer
233 views

A subadditive maximal ergodic theorem

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $\tau:\Omega\to\Omega$ be a measurable map on $(\Omega,\mathcal A)$ with $\operatorname P\circ\:\tau^{-1}=\operatorname P$, $Y_n:\...
0xbadf00d's user avatar
  • 167
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 ...
cantorhead's user avatar
2 votes
1 answer
1k 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 ...
2 votes
1 answer
159 views

Can we show that this transition semigroup preserves a certain Wasserstein space?

Let $E$ be a separable $\mathbb R$-Banach space, $v:E\to[1,\infty)$ be continuous, $$\rho(x,y):=\inf_{\substack{\gamma\:\in\:C^1([0,\:1],\:E)\\ \gamma(0)\:=\:x\\ \gamma(1)\:=\:y}}\int_0^1v\left(\gamma(...
0xbadf00d's user avatar
  • 167
2 votes
1 answer
1k views

Understanding measure-preserving transformation [closed]

Given measure space $(S, \mathcal{S}, \mu)$, and measurable function $\phi: S \to S$. $\phi$ is measure-preserving if $\forall A \in \mathcal{S}, \mu(A) = \mu(\phi^{-1}(A))$. My confusion is that why ...
Jokerr's user avatar
  • 23
2 votes
1 answer
200 views

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 ...
Landon Carter's user avatar
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 ...
David Berman's user avatar
2 votes
1 answer
119 views

time delay ergodic theorem

given dynamic system $(X, \mathcal{B}, F, \mu), \mu \circ F^{-1}=\mu, F $ is mixing, $ A \in \mathcal{B}, s.t. \mu(A) >0 $. consider dynamic system $(X\times X, \mathcal{B}\otimes \mathcal{B}, ...
jason's user avatar
  • 553
2 votes
1 answer
122 views

How is this bound for a Wasserstein contraction coefficient in this paper obtained?

I'm trying to understand the following conclusion from this paper (see below for the relevant paragraphs): I'm not sure whether they really mean that it follows from the statements of Lemma 3.2 (...
0xbadf00d's user avatar
  • 167
2 votes
0 answers
92 views

Existence of ergodic subgroup invariant to a product measure

Let $X=\{0, 1\}^{\mathbb{N}}$ and $G$ be the group of permutations, each of which only permutes finitely many coordinates of $X$. Fix a sequence $(\lambda_n)_{n\in \mathbb{N}} \subseteq (0, 1]$ and ...
Sanae Kochiya's user avatar
2 votes
2 answers
328 views

Existence of the limit of periodic measures

Let $T: X \to X$ be a continuous map over a compact metric space. We say that a measure $\mu$ is $T$-invariant if $T_{\ast} \mu= \mu$. We denote by $M(X, T)$ the space of all $T$-invariant Borel ...
Adam's user avatar
  • 1,043
2 votes
0 answers
123 views

Probability of a finite cylinder set in a free group

Let $\mathbb{F}_n$ be the free group (each elemen is in its reduced form) generated by the set $\Sigma_n = \{a_1, a_2, \cdots, a_n, a_1^{-1}, a_2^{-1}, \cdots, a_n^{-1}\}$ and let $e$ denote the ...
Sanae Kochiya's user avatar
2 votes
0 answers
117 views

Estimating the measure of a pre-image of a polynomial

This question was previously posted on MSE https://math.stackexchange.com/questions/3305781/estimating-the-measure-of-a-pre-image-of-a-polynomial Let $\sigma := 2/(3\sqrt{3})$, be a real number. And ...
Matheus Manzatto's user avatar
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 $\mathrm{avg}_{f,n}(x)$...
Monroe Eskew's user avatar
  • 18.6k
1 vote
1 answer
189 views

If a Markov semigroup is eventually contractive, can we conclude that it admits a unique invariant measure?

Let $E$ be a separable $\mathbb R$-Banach space, $\rho$ be a complete separable metric on $E$, $\operatorname W_\rho$ denote the Wasserstein metric of order $1$ associated to $\rho$, $\mathcal M_1(E)$ ...
0xbadf00d's user avatar
  • 167
1 vote
1 answer
183 views

If $(κ_t)$ is a semigroup with invariant measure $\mu$ and $ν$ is singular to $\mu$, then $νκ_t$ might not converge to $\mu$ in total variation norm

Let $E$ be a Polish space, $(\kappa_t)_{t\ge0}$ be a Markov semigroup on $(E,\mathcal B(E))$, $\mu$ be a probability measure on $(E,\mathcal B(E))$ invariant with respect to $(\kappa_t)_{t\ge0}$ and $\...
0xbadf00d's user avatar
  • 167
1 vote
0 answers
56 views

Minimizing the rate of geometric ergodicity of a Metropolis-Hastings kernel depending on a parameter

Let $\tilde\kappa$ denote the transition kernel of the Markov chain generated by the Metropolis-Hastings algorithm with proposal kernel $\tilde Q$ and target distribution $\tilde\mu$. I want to ...
0xbadf00d's user avatar
  • 167
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-* ...
Bruno Brogni Uggioni's user avatar
0 votes
2 answers
223 views

Induced probability measure on a finite orbit under a group action

Suppose we have a discrete group $G$ acting on a compact set $X \subseteq \mathbb{R}^d$ via measure-preserving homeomorphisms, and suppose we have a point $x$ whose orbit $Gx$ is finite (say $|Gx| = n$...
James Propp's user avatar
  • 19.7k
0 votes
1 answer
79 views

Reduce ergodicity to the ergodicity of the coordinate process

Let $(E,\mathcal E,\lambda)$ be a probability space and $\lambda$ be a measurable map on $(E,\mathcal E)$ with $\lambda\circ\tau^{-1}=\lambda$. I would like to show that $\tau$ is $\lambda$-ergodic ...
0xbadf00d's user avatar
  • 167