# Questions tagged [ergodic-theory]

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.

1,065 questions
Filter by
Sorted by
Tagged with
11 views

### An ergodic stationary stochastic process

Can you give me an example of discrete-time stochastic process on $\mathbb{R}_+$ that is stationary and ergodic but does not have independence in time?
15 views

41 views

121 views

49 views

### Which probability inequality is applied in this proof?

I'm trying to understand which probability inequality is used at the end of the following proof of Kingman's theorem in Revuz' Markov Chains book. He's considering a probability space with probability ...
27 views

### How is ergodic measure defined?

I have a clear definition for an ergodic endomorphism, but I can't seem to find a clear definition for an ergodic measure. For example, the Wolfram definition (https://mathworld.wolfram.com/...
71 views

52 views

### Finite-to-one measurable factor map preserves entropy?

Let $X$ and $Y$ be compact metric spaces with Borel $\sigma$-algebras $\mathcal{B}$ and $\mathcal{C}$. Let $S: X \to X$ and $T: Y \to Y$ be homeomorphisms. Let $\pi: X \to Y$ be a bounded-to-one ...
66 views

### If $T$ is measure preserving, then $f\mapsto f\circ T$ is an isometry on $L^\infty$

Let $(E,\mathcal E,\mu)$ be a probability space and $T:E\to E$ be $(\mathcal E,\mathcal E)$-measurable with $$T_\ast\mu=\mu\tag1.$$ How can we show that $$\mathcal L^p(\mu)\ni f\mapsto f\circ T\tag2$$ ...
54 views

### Proof of source coding theorem using asymptotic equipartition property(AEP)

In this wikipedia article, there is a proof given for one of the directions of the Shannon's source coding theorem using the asymptotic equipartition property (AEP). I am unable to follow the proof. ...
73 views

### What is the measure of the function?

Is the $T$-function $f(x) = (x \text{AND} c) + (x^2 \text{OR} c)$, where $c$ is positive integer ergodic in the space $Z_2$ (p-adic numbers)? What is the measure of this function? I am trying to use ...
125 views

### Is this a property of the sine function?

Because of the equidistribution property of $\{n\mod{2\pi}\}_{n\in\mathbb{Z}}$ in $[0,2\pi]$, my intuition tells me that the following statement about the sine function must be true. We can find an ...
87 views

### Mean ergodic theorem different assumption

Show that the mean ergodic theorem still holds if we replace the assumption that $T$ is an isometry by the assumption that $T$ is contraction, that is $||T f|| ≤ ||f||$ for all $f ∈ H$ where $H$ is ...
40 views

### Space of ergodic measures is $G_\delta$ in space of invariant measures

I'm reading an old paper by Varadarajan (1963) titled "Groups of automorphisms of Borel spaces" and I'm trying to fulfill the details. There are many questions I thought about for quite a long time ...
36 views

### Understanding of Random Variables generated by Measure Preserving Transformations

I have difficulties to understand this in ergodic theory (approach form proabrbility theory). Usually, an ergodic process is defined by: $$X_n(\omega) := X(\phi^n(\omega)),$$ where $\phi$ is a ...
16 views

### Formalizing Siegel lattice integral formula

Siegel's lattice integral formula says that for $f \in L^1 (\mathbb R^n-[{0}])$, $$\int_{\mathbb R^n} f(x) dx = \int_{L_n} f^{*}(\lambda)$$ where on the right we integrate over the covolume lattices ...
130 views

### Birkhoff averages convergence

Let be $(X,\mathcal{A},\mu)$ a probability space, $T:X\to X$ a measurable tranformation preserving $\mu$ and $f: X \rightarrow \mathbb C$ a measurable function. Show that for almost every $x \in X$ ...