# Tagged Questions

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### For a non-compact metric space, do I have that the set of $\sigma$-invariant measures is compact?

Let $X$ be a non-compact metric space with a sub shift $\sigma: X \to X$. Do I have that the the space of $\sigma$-invariant probability measures on $X$ such that $\mu (B) = \mu (\sigma^{-1}(B))$ with ...
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### Is Kolmogorov's zero–one law undecidable?

Kolmogorov's zero–one law is related to other parts of probability like the law of large numbers. However it is stated that what the actual probability of a tail event is (either 0 or 1) is hard to ...
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### On Ergodicity of the product of two Ergodic transformations

I was recently reading some textbooks and topics in Ergodic theory that I found this fascinating result which I couldn't prove completely: Suppose that $T$ and $S$ are two Ergodic transformations. ...
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### Ergodic Theory, Bernoulli Measure, Cylinder Set

Let $N\geq2$ be an integer and consider the probability space ($\Sigma^+$,B, $\mu_p$) where $\mu_p$ is the Bernoulli measure with respect to probability vector $\ p = (p_1,...,p_N)$ . Show that for ...
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### A variation of Birkhoff's Ergodic theorem

Suppose $T_k$ are measure perserving for each $k$, and $f$ is integrable. Is it true that $\frac{1}{n}\sum_{k=1}^n f(T_k\circ T_{k-1}\dots\circ T_1 \omega)$ converge almost surely?
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### What is “abstract” ergodic theory?

This is just a question about the usage of the term "abstract". What kind of questions in ergodic theory is considered "abstract" and what's a "regular" question? From some seminars it seems that ...
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### What keeps measure-preserving transformations from concentrating in a particular portion of a probability space?

I'm trying to show that for an event A with positive probability there is some n bounded by 1/P(A) such that $P(A \cap$ T$^{-n}A) > 0$, where T is a probability-preserving transformation. I'm ...
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### Ergodic Process: Does it visit all state?

I read in this article: " Ludwig Boltzmann, coined "ergodic" as the name for a stronger but related property: starting from a random point in state space, orbits will typically pass through every ...
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### simple proof of the $L^2$ weak law for discrete-time ergodic Markov processes

Let $\{X_t\}_{t\in\mathbb{Z}}$ be a stationary and ergodic stochastic process with finite second moment. Von Neuman's ergodic theorem implies that the time average $(1/N)\sum_{j=0}^{N-1} X_j$ ...
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### What is the right invariant $\sigma$-algebra for the Birkhoff ergodic theorem?

I have been reading stuff about ergodic theory, and I have encountered two versions of the involved "invariant sigma field". let the underlying probability space be $(\Omega,\mathcal{F},P)$, and let's ...
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### Proving Borel Cantelli Lemma using Martingales

I need a hint for exercise 5.2.1 in the book: Ergodic Theory: with a view towards Number Theory By Manfred Leopold Einsiedler, Thomas Ward. In the chapter 5 the authors gives the margingale ...
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### literature to learn more on ergodic harris recurrent chains with an atom

I'm trying to learn more on the topic mentioned in the title. Namely I'd like to get more information on the behavior of the boundary terms. ie if I decompose sum of my chain (suppose it's ...
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### 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. ...
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### Measurable Partition and Ergodic Decomposition

I need some background before asking the question: Let $\mathcal{P}$ is called a measurable partition if there is a measurable set $M_0\subset M$ with full probability measure such that, restric to ...
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### stationary and ergodic

Let $p\in \mathbb N$ and $a_1,\ldots,a_p\in \mathbb R$. Denote by $x$ the sequence $$x=(x_k)_{k\in\mathbb N }=(a_{k \bmod p})_{k\in\mathbb N }$$ where $(k \bmod p)\in{1,\ldots,p}$ is the remainder ...
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### 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% ...
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### convergence of entropy and sigma-fields

This question is related to this one. Let $(X_1, X_2, \ldots)$ be a sequence of random variables such that each $X_n$ takes its values in a finite space, say $\{0,1\}$, and the $\sigma$-field ...
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### Pinsker $\sigma$ Algebra

Let $(X,A,\nu)$ be a probability space and $T:X\to X$ a measure-preserving transformation. The Pinsker sigma algebra is defined as the lower sigma algebra that contains all partition P of measurable ...
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### The prime numbers do not satisfies Benford's law

A set of numbers is said to satisfy Benford's law if the leading digit d (d ∈ {1, ..., 9}) occurs with probability, $$P(d)=\log_{10}(d+1)-\log_{10}d,$$ how do you prove that the prime numbers do ...
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A set of numbers is said to satisfy Benford's law if the leading digit d (d ∈ {1, ..., 9}) occurs with probability, $$P(d)=\log_{10}(d+1)-\log_{10}d$$ With Birkhoff's Ergodic Theorem is possible ...
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### Generalization of Dobrushin's Ergodic Decomposition for continuous Markov Chains

Let $T$ be the shift transformation. Let $P$ be invariant for $T$ and also define a discrete state space Markov Chain. Let $C_{1},\ldots,C_{n}$ be the connected components of the Markov Chain. It ...
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### How follows the Strong Law of Large Numbers from Birkhoff's Ergodic Theorem?

We want to prove the strong law of large numbers with Birkhoff's ergodic theorem. Let $X_k$ be an i.i.d. sequence of $\mathcal{L}^1$ random variables. This is a stochastic process with ...
Let $(\Omega, {\cal F},P)$ be a complete probability space and $T$ a mesure-preserving transformation on $\Omega$ that is ergodic. The point-wise ergodic theorem states that for any $f\in L^1(P)$, ...
Consider a Markov chain, for simplicity let us consider time discrete chains. The problem We consider a TDMC (Time Discrete Markov Chain) $(X_t)_{t \geq 0}$ with $X \in \mathcal{X}$ (having ...