0
votes
0answers
32 views

Why define shift invariant set as $\Lambda=\tau^{-1}\Lambda$?

Can we define shift invariant set as $\Lambda = \tau\Lambda$ instead of $\Lambda = \tau^{-1}\Lambda$, where $\tau$ is the shift operator? Can permutable set be defined as either $\Lambda = \pi ...
0
votes
1answer
29 views

ergodic theorem in the seasonal component analysis of time series

When studying the "seasonal components" part of time series, I once read the following statement. I do not understand what role does the ergodic theorem play here? The decomposition of the process ...
4
votes
1answer
76 views

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 ...
2
votes
0answers
70 views

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 $(1/N)\sum_{j=0}^{N-1} X_j$ ...
3
votes
2answers
90 views

Ergodic for the mean, but not ergodic stochastic process?

Is there an example of a strictly stationary (zero mean, finite variance) stochastic process $(X_t\mid t\in \mathbb{N})$ that satisfies the conclusion of the ergodic theorem, i.e., the sample mean ...
1
vote
0answers
70 views

Different definitions of an ergodic stationary process

From page 3 of a note: A stationary process is ergodic if any two variables positioned far apart in the sequence are almost independently distributed. A formal definition is the following: ...
4
votes
2answers
93 views

Optimal probability measure

Let $A$ be a finite set and let $\Bbb P$ be a probability measure on $A^{\Bbb N_0}$. Further, let $x_i:A^{\Bbb N_0}\to A$ be projection maps, so that $(x_i)_{i=0}^\infty$ can be treated as a ...
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 ...
3
votes
2answers
336 views

Examples: invariant events

In a couple of books I'm reading chapters devoted to the Ergodic theory. As a setting: $(\Omega,\mathcal F,\mathsf P)$ is a probability space, $X:(\Omega,\mathcal F)\to (S,\mathcal S)$ is random ...