Question about probability spaces $(X,\mathcal B,\mu)$ with a measurable map $T\colon X\to X$ preserving the measure, that is $\mu(T^{—1}A)=\mu(A)$ for all $A$ measurable.

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24 views

Converges of measures.

Good afternoon, we have the following: Let $(Y,d)$ is a general metric space, $\mathcal{M}(Y)$ is the set of finite Borel measures on $Y$ and $C_B(Y)$ denotes the Banach space of bounded continuous ...
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1answer
65 views

A Question Regarding Markov Chains and Ergodicity

Suppose the Markov chain with Probability Transition Matrix, $P$ = ($p{_x}{_y}$) is ergodic and $p{_m}(x, y) > 0$ for all states $x$ and $y$. If $n ≥ m$, show that $p_n(x, y) > 0$ for all ...
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1answer
26 views

Approximate eigenvalues of an ergodic invertible transformation

Consider a non-atomic probability space $(X,\mathcal{B}, m)$. Let $T: X \to X $ be an ergodic invertible measure preserving transformation.Let $U_T$ be the Koopman operator associated with $T$. Show ...
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34 views

Ergodic actions of orthogonal group $O(2)$

I am looking for explicit ergodic action of $O(2)$ on a von Neumann algebra $M$. ($O(2)$=orthogonal group of $2\times 2$ matrix)
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1answer
48 views

application of Birkhoff Ergodic Theorem

Let $(\Omega, \mathcal{F} , P)$ be a probability space and let $T$ and $S$ be ergodic, measure-preserving transformations of $(\Omega, \mathcal{F} , P)$. Let $X : \Omega → \mathbb{R}$ be a bounded ...
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1answer
48 views

A question of determining when the entropy is maximum.

Y ={ 1, 2,...,r} We are given that X is the set of two sided sequences with entries from Y and T is the two sided shift on X, and m is a T invariant probability measure on X. If $p_i = m(\{x ...
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0answers
18 views

Ergodic measure for action of $SO_2$ on lattice

Let $X:= \Gamma/PSL_2(\Bbb R)$ and for each $x \in X$ define $\phi_x(g):= xg^{-1}$ for $g \in SO_2$. Then the induced measure $(\phi_x)_*m_{SO_2}$ is ergodic for the $SO_2$ action and is a factor of ...
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1answer
36 views

Non-atomic, ergodic measure which is left and right shift invariant.

Given a one-sided shift space, say $X = \prod\limits_{n=1}^\infty \mathbb Z_2$. Denote the left shift by $T$: $T(x_1 x_2 x_3\cdots) = x_2x_3 \cdots$. There are lots of examples of $T$-invariant ...
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1answer
27 views

Are $L_\infty$ functions measurable/integrable?

Lemma 2.6 of "Ergodic Theory with a view towards Number Theory" (Einsiedler-Ward) involves: $$ \int f d\mu $$ where $f \in L^{\infty}$. Actually it is a calligraphic $L$ and I'd love if you would ...
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1answer
36 views

What is known about the space of measure-preserving transformations?

I started reading about measure-preserving transformations, the ergodic theorems and mixing, but I was also wondering what is known about the space of measure-preserving transformations. The books ...
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1answer
42 views

ergodicity in $\mathbb{Z}^d$

Fix $d \geq 1$ and let $E(\mathbb{Z}^d)$ denote the set of all edges of the graph $\mathbb{Z}^d$. Let us consider a measure preserving system $(\mathbb{R}^{E(\mathbb{Z}^d)}, ...
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2answers
24 views

measure preserving system

Let $T$ be a measure-preserving transformation on a probability space $(\Omega, \mathcal{F}, P)$ and let $A \in \mathcal{F}$ such that $P(A) > 0$. (i) Show that there exists $n \geq 1 $such that ...
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1answer
21 views

Measure preserving ergodic map commutes with complementation?

This is probably trivial (in which case I apologize), but it's late and I would really like a quick proof/counterexample for this (for a different problem that I'm doing): if $(X,\mathcal{M},\mu,T)$ ...
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1answer
17 views

Initial point and initial distribution of the Markov chains

I am reading about Markov chains on a general state space and the ergodicity theory. Some of the ergodic theorems are presented when we consider n-step transition probability conditional on initial ...
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1answer
33 views

Why is the shift map ergodic?

Given a finite set $S$, in the space of strings $\Sigma=S^{(\omega)}$ equipped with the Bernoulli measure $\mu$, I want to know why the shift map $\sigma:\Sigma\rightarrow \Sigma $, define as ...
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2answers
24 views

Simple eigenvalue of Koopman operator

Let $T : X \to X$ be a measure-preserving transformation and $U_T : L^2(X, \mu) \to L^2(X, \mu)$ , $(U_T f) (x) = f(Tx).$ What does it mean a $\bf{simple}$ eigenvalue of $U_T$? $\lambda \in ...
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14 views

If $1$ is a simple eigenvalue of $T$, then $T$ is an ergodic measure-preserving transformation

Let $T : X \to X$ be a measure-preserving transformaton and $U_T : L^2 (X, \mu) \to (X, \mu)$, $$(U_T f) (x) = f(Tx).$$ I have to show that if $1$ is a simple eigenvalue of $U_T$, then $T$ is ...
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1answer
21 views

Different ergodic probability measures are mutually singular

Can someone, please, give me a hint on how to demonstrate that different ergodic probability measures are mutually singular? Thank you!
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1answer
54 views

Ergodicity of stochastic process

If one can show that the process converges to a stationary process in probability, does it mean that the process is ergodic?
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1answer
27 views

Example for a non-ergodic stationary process

Let $(X_n)_{n \in \mathbb{N}}$ be a (strictly) stationary process and let $T$ denote the left-shift on $\mathbb{R}^\mathbb{N}$, i.e. $T((x_n)_{n \in \mathbb{N}}) = (x_{n + 1})_{n \in \mathbb{N}}$. ...
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15 views

Theorem of Daniell-Stone (uniqueness when assuming compactness)

Theorem of Daniell-Stone. Let $L$ be a $\sigma$-continuous abstract integral on a Stone lattice V of real-valued functions on $\Omega$ and let $\mathcal{A}(V)$ denote the set of all $V$-open ...
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1answer
34 views

Strengthening Poincaré Recurrence

Let $(X, B, \mu, T)$ be a measure preserving system. For any set $B$ of positive measure, $E = (n \in \Bbb N |\; \mu(B \; \cap \;T^{-n}B) > 0)$ is syndetic. This exercise comes from Einseidler ...
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0answers
65 views

Ergodicity of stochastic recursive process

Does anyone know how one can show ergodicity for a recursive stochastic process determined by the following equation: \begin{equation} X_n = f(\varepsilon_{n-1},Y_{n-1})X_{n-1}, ...
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1answer
21 views

Criterium in ergodic theory.

Given a topological space $X$ with a probability measure $\mu$ and a continuous transformation $T:X \rightarrow X$ which preserve measure. If a set $A$ with $1>\mu(A)>0$ is such that the ...
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1answer
35 views

Approximating Borel Measure with Atomic Measures

I see some posts that are related to this one, e.g. Borel Measures: Atoms (Summary) I have a sort of particular question: I have one professor saying the following is true, while another says it's ...
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1answer
36 views

Connection between Ergodic Theory and Markov Chains

Could someone suggest a good reference where the connection between Ergodic Theory and (ergodic) Markov Chains is nicely explained ?
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1answer
68 views

Does the Central Limit Theorem Imply the strong Law of Large Numbers?

Assume that $(X_{k})_{k\geq 0}$ is a stationary (or weakly stationary) process defined on the same probability space $(\Omega,\mathcal{F},\mathbb{P})$. Can we assert from the convergence in ...
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0answers
47 views

Ergodic Theorem for Markov Chain with one closed communicating class and several transient states

It is known, that if a markov chain with a finite state space has only one closed aperiodic communicating class and several transient states, then there is a unique stationary distribution for this ...
3
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0answers
33 views

Writing a Hilbert C*-submodule of $L^2(X)$ as an integral sum over Hilbert subundles

In Ergodic Theory, some (though not all) presentations of compact extensions use Hilbert bundles. Given $(X,\mathcal{B},\mu,T)$ and a sub $\sigma$-algebra $\mathcal{G}\subseteq\mathcal{A}$ one has an ...
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1answer
22 views

Showing the ergodicity of a rotation on the unit sphere

Consider the rotation $R_\alpha(z) = \alpha z, R_\alpha : S^1 \to S^1$. Show that $R_\alpha$is ergodic with respect to Haar measure on $S^1$ $\iff$ $\alpha$ is not a root of unity. I don't know how ...
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1answer
56 views

Ergodicity vs Weak Mixing

I have been trying to prove the following: Let $(\Omega,\mathcal{F})$ be a measurable space endowed with probability measure $\mathbb{P}$. Suppose $\tau : \Omega \to \Omega$ is a measure preserving ...
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1answer
27 views

Birkhoff Ergodic theorem for two measures

Suppose $(X,\mathcal{B}, \mu, T)$ and $(X,\mathcal{B}, \nu, T)$ are both ergodic ppt. I'm a bit confused how the B Ergodic Theroem works since the LHS of the equation doesn't involve $\mu$ or $\nu$, ...
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1answer
25 views

Wandering set definition

I've seen two apparently different definitions and am wondering which is correct. A set $W$ is wandering if $\{T^{-k}W; k\in \mathbb{N}_0\}$ (resp. $\{T^{k}W; k\in \mathbb{N}_0\}$) are pairwise ...
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33 views

Law of large numbers extension

I have a process $\{X_t\}_{t\ge0}$ which is not iid. All the $X_t$ have finite first moment and via simulation I have a strong feeling that $$ \lim_{m\rightarrow\infty}\frac{1}{m}\sum_{t=0}^{m-1}X_t = ...
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0answers
183 views

What's the probability distribution of a deterministic signal or how to marginalize dynamical systems? (functional integrals in probability theory)

In many signal processing calculations, the (prior) probability distribution of the theoretical signal (not the signal + noise) is required. In random signal theory, this distribution is typically a ...
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1answer
24 views

Using Furstenberg's skew-product for $\alpha n^2$ equidistributed.

I am having trouble proving that if $\alpha_1,\alpha_2$ are irrational, then the sequence $(\alpha_1n,\alpha_2n^2)$ is equidistributed in $\mathbb{T}^2$. It is straightforward to use Furstenberg's ...
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1answer
26 views

Approximation of integrals smooth functions via sampling with non-Liouville number.

Suppose $\alpha$ irrational is not a Liouville number. i.e. we cannot find an a sequence of rational approximations $p_n/q_n$ with $|\frac{p_n}{q_n}-\alpha|<\frac{1}{q_n^n}$. I am trying an ...
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2answers
26 views

Consequences of Arnoux, Ornstein, Weiss Theorem.

The theorem states that any invertible, aperiodic, measure-preserving system on a Borel probability space is isomorphic to a cutting and stacking transformation. My question is, why is this useful? ...
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1answer
30 views

Rotation of the torus $T^2$ by irrational numbers linearly dependent over $\mathbb Z$

It is known that the rotation $x \to x + \alpha$ of $S^1 = \mathbb R / \mathbb Z$ with irrational $\alpha$ is ergodic and, in particular, $\alpha n$, $n = 1, 2,\dots$, are dense in $S^1$. In two ...
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2answers
102 views

Initial conditions that converge to an unstable equilibrium

Consider the discrete-time dynamical system \begin{align} x_{k+1}=T(x_k), \end{align} where $k\in\mathbb{N}$, $x_k\in\mathbb{R}^n$, $x_0$ is the initial condition, and $T:\mathbb{R}^n\to\mathbb{R}^n$ ...
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0answers
28 views

Spectrum of $T^2$

Is it possible that $T^2$ has a discrete spectrum when $T$ is an invertible measure-preserving transformation whose spectrum is continuous or mixed ?
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1answer
21 views

can two Markov kernels be close in total variation and differ in their ergodicity properties?

This is a question inspired by a recent MCMC paper and linked to an earlier question of Sam Livingstone that did not get any answer. Given two Markov kernels $\mathfrak{K}$ and $\mathfrak{H}$ such ...
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1answer
33 views

Ergodicity of the natural measure implies uniqueness of the invariant density?

Consider a dynamical system $x_{t+1} = F(x_t)$ defined in $\Omega$ and its natural measure to be $\mu$. The Perron-Frobenius operator $F$ maps the density $f$ in time according to $f_{t+1} = F(f_t)$. ...
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25 views

Why does ergodicity not neccesarily imply ergodic for the mean?

I'm trying to answer a question where I have an ergodic and covariance stationary process $\{x_t\}$, and without imposing further moment conditions need to prove $\frac{1}{n} \sum\limits_{t=1}^n x_t^2 ...
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1answer
51 views

Ultrafilter and upper natural densities

It is straightforward to show that there is an ultrafilter $\mathcal{U}_0$ on the positive integers such that every element $A\in \mathcal{U}_0$ satisfies $$d^\ast(A):=\limsup_{n\to +\infty} ...
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2answers
92 views

An intuitive explanation of how the mathematical definition of ergodicity implies the layman's interpretation 'all microstates are equally likely'.

I'm self-studying Statistical Mechanics; in it I got Fundamental Postulate of Statistical Mechanics and that took me to ergodic hypothesis. In the most layman's language, it says: In an isolated ...
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0answers
32 views

Moving Average of an Ergodic Markov processes.

Let $ \{X(t); t\geq 0\} $ be an ergodic Markov process, and let G be a positive integrable function with $\int_0^\infty G(x)dx=1$. Does $$F(t)\doteq G*X(t) = \int_0^t G(t-s)X(s)ds $$ converge in ...
3
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107 views

Zeta functions for flows on hyperbolic surfaces

The Riemann zeta function is defined by $$ \zeta(s)=\sum_{n=1}^\infty \frac1{n^s} $$ and can be written in the form $$ \zeta(s)=\prod_p\frac1{1-p^{-s}}, $$ where the product is over all prime numbers ...
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26 views

Question about a type of continuous state Markov-process.

EDIT: Solved! It turns out that if the function is continuous and various regularity conditions hold then the statement is true. This has been established in the 'stochastic approximation' literature, ...
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1answer
33 views

multiples of subset of $\mathbb{N}$

Suppose that $ A_1\cup \dotsm A_n=\mathbb{N}$ is a partition of $\mathbb{N}$ into disjoint subsets. Is it true that there is an integer $1 \leq k \leq n$ such that the set $A_k\cap 2A_k$ is infinite?