# 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.

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### Ergodicity when the variance doesn't converge

A power law (x^(-k)) isn't ergodic for k<= 2 given that the expected value doesn't exist (nothing to converge to). However for 2<k<=3 there is a finite expected value but an infinite variance....
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### Is the translation on torus $\mathbb{T}^2$ ergodic with the haar measure?

Let $f: \mathbb{S}^1 \times \mathbb{S}^1 \to \mathbb{S}^1 \times \mathbb{S}^1$ $f(x,y)=(cx,y)$ such that $c$ is a complex number and $|c|=1$. Consider the haar measure. Is it an ergodic system? Why? I ...
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### Sufficient condition for the convergence of the time average of a dynamical system

Consider a dynamical system defined by a 1-d unimodal map. Suppose that the topological entropy of the system is positive. Then does the time average of f converge to a constant? If so, give me any ...
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### A special case of Birkhoff's ergodic theorem for a nonexpansive map

I have the following difference equation: $x_{t+1}=f(x_t)=x_t+3.61*(1/x_t-1)$ where the domain is the closed interval E=[0.19,17.48]. I know the following facts for this map. This map is smooth and ...
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### Application of ergodic theorem to transformed Gaussian process

Assume that $W$ is a zero-mean Gaussian process on $[0,\infty)$ with covariance function satisfying $c(s,t):=E(W(s)W(T))=C(|t-s|)$ for a function $C:[0,\infty) \to \mathbb{R}$. Can I exploit somehow ...
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### A maximal invariant generates the invariant $\sigma$-algebra

Let $G$ be a group action on the set $X$, and let $f:(X,\Sigma_X)\to (Y,\Sigma_Y)$ be a measurable maximal invariant ($f$ is constant on each orbit and take different values on different orbits). Am ...
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### When do mixtures of ergodic Markov kernels remain ergodic?

Given two Markov kernels on the same space $\mathfrak X$ and relative to the same dominating measure, $K_0(\cdot,\cdot)$ and $K_1(\cdot,\cdot)$, both ergodic with respective stationary distribution ...
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### Oddity in definition(s) of quasi compact operator

I was wondering about the general definition of a quasicompact operator. There seem to be two main ones floating around in the literature, and I am not sure they are equivalent. The first, and ...
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### Uniform law of large numbers for ergodic stationary sequence

I am trying to apply a uniform law of large numbers, which is stated in Lemma 7.2 of "Econometrics" by Fumio Hayashi. The starting point is the stochastic process $\{x_t\}$, which we assume ...
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I'm studying the entropy for flows via reparametrizations. Let (X,d) a compact metric space. For a closed interval $I$ which contains the origin, a continuous map $\alpha: I \rightarrow \mathbb{R}$ is ...
Let $\Sigma=\{1,2\}^{\mathbb{N}}$ and $A$ be the sigma-algebra generated by the cylinders sets $\{w∈Ω|∀s∈S,w_s=ϵ_s\}$ with $S⊂\mathbb{N}$ finite and $ϵ_s∈\{0,1\}$ . Let $p∈(0,1)$ . We take product ...