2
votes
0answers
28 views

Geometric ergodicity and mixing - stationary case

I have this theorem: The Markov Chain {$X_n$} is stationary and geometrically ergodic if and only if {$X_n$} is stationary and absolutely regular with $\beta_n=O(\gamma^n)$ for some $\gamma\in ...
0
votes
0answers
17 views

a sequence if ergodic iff the limit of (f(X0)g(X0)+f(X1)g(X0)+…+f(Xn-1))/n as n goes to infinity equals to Ef(X0)Eg(X0).

Let X0,X1,......be a stationary sequence and denote the distribution of X0 by D.Show that the sequence is ergodic if and only if for every f,g in L2(D) the limit of ...
1
vote
1answer
57 views

A random variable, measurable w.r.t. the class of invariant events, is invariant

I would like to deeply understand the following proof; however, many things are not really obvious to me. I would appreciate if somebody could explain the solution in detail. Thank you. Problem ...
1
vote
0answers
71 views

Conclusions about the Manneville-Pomeau Example

I was trying to solve an exercise about the Manneville-Pomeau example, but I got stuck. I need to give a little background before posting the exercise. Thank you guys in advance for your attention. ...
2
votes
2answers
59 views

Convergence of an Ergodic process

I'm having trouble working through the math of the problem below. I believe the problem as described is an ergodic process. I've written a simple simulation of the problem, that converges to 66.6...6% ...
14
votes
1answer
459 views

Kakutani skyscraper is infinite

Karl E. Petersen's book "Ergodic Theory", chapter 2, exercise 9, on page 56 Prove that for any ergodic measure preserving transformation $T:X\rightarrow X$ on a non-atomic probability space $(X, ...
2
votes
0answers
74 views

Ergodic/stochastic convergence

I do have a problem with my homework, and to be honest I am simply lacking any idea on how to begin- maybe someone could give me a tip. First off, here is the assignment: The whole assignment deals ...
1
vote
1answer
67 views

How can I improve this proof? On $\sum\limits^\infty f(T^nx)=\infty$ for a.a. $x \in \{f >0\}$.

Let $(X,\mathcal{F},\mu)$ be a prob. space and $T:X \rightarrow X$ be measure preserving, and $f \in \mathcal{L}^1(X,\mathcal{F},\mu)$. Assume that $f \geq 0$ and suppose the set $A = \{f > 0\}$ ...
0
votes
1answer
75 views

Trying to show an implication for a measure preserving $T$ on a probability space.

Here's the problem: Let $(X,\mathcal{F},\mu)$ be a probability space and $T:X \rightarrow X$ a measure preserving transformation. Show that if $f(Tx) \leq f(x)$ $\mu$ a.e., then its holds that $f(Tx) ...
0
votes
1answer
78 views

The first return time of an irrational rotation

For the probability space $([0,1),\mathcal{B},\lambda)$ and for an irrational $\theta \in (0,1)$ we have the map $Tx = x + \theta \bmod 1$. I'm trying to find an expression for $n(x):=\inf\{n \in ...
10
votes
1answer
288 views

$(n^2 \alpha \bmod 1)$ is equidistributed in $\mathbb{T}^2$ if $\alpha \in \mathbb{R} \setminus \mathbb{Q}$

I did the following homework question, can you tell me if I have it right? We want to show that the sequence $(n^2 \alpha \bmod 1)$ is equidistributed if $\alpha \in \mathbb{R} \setminus \mathbb{Q}$. ...
1
vote
0answers
80 views

To construct an invariant measure on the product space

$M$ is a separable complete metric space. $N$ is a compact metric space. $F$, defined by $F(x,y)=(f(x),g(x,y))$, is a continuous transformation from $M\times N$ to itself. We know there is a measure ...
0
votes
1answer
84 views

Forward orbits and backwards orbits same closures

Assume that $T: X \to X$ is a homeomorphism of a compact metric space to itself, and let $\mu$ be a $T$-invariant Borel probability measure on $X$. I want to show that for almost every $x \in X$ ...
2
votes
1answer
169 views

A generic point for a non-ergodic measure

Definition: Let $X$ be a compact metric space and let $\mu$ be a Borel probability measure on $X$, then we say that the sequence $(x_n)$ of elements of $X$ is equidistributed with respect to $\mu$ ...