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Results tagged with ergodic-theory
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user 12978
Dynamical systems on measure spaces, invariant measures, ergodic averages, mixing properties.
1
vote
Accepted
Is there a way to find $\limsup$ and $\liminf$ for ergodic processes almost surely?
For ergodic $X_n$, almost surely
$$\limsup_n f(X_n) = \sup_n f(X_n) = \sup \{a \in \mathbb{R} : Pr(f(X_0) > a) > 0\}.$$
In other words, by recurrence, what can happen (with positive probability) wil …
- 6,287
9
votes
2
answers
745
views
Is this ergodic inequality true?
Is anything similar to the following inequality true,
$\displaystyle P\{\max_{n \leq k \leq m} |A_k f - A_n f| > \epsilon\} \leq C \frac{||A_m f - A_n f||_1}{\epsilon}$
where $A_n f = \frac{\sum …
- 6,287
11
votes
2
answers
2k
views
De Finetti's theorem, the pointwise ergodic theorem, and reverse martingales
De Finetti's theorem says that an exchangeable sequence of random variables $X_i$ is a mixture of i.i.d. random variables. In other words, if $\mu$ is a measure on $\mathbb{R}^\infty$ that is invaria …
- 6,287
5
votes
De Finetti's theorem, the pointwise ergodic theorem, and reverse martingales
(My understanding of this material has significantly gone up in the months since I asked it, and I will attempt to answer my own question.)
In general, if $(\Omega,\mathcal{B},\mathbb{P},\{T_g\})$ is …
- 6,287