<p>I' ve come across two notions for the condition of (mean-) ergodicity and wonder whether and how they relate.</p>
<p>(1) Let <span class="math-container">$ {X}_{t \ge 0} $</span> be a second order stationary process with (constant) mean <span class="math-container">$\mu$</span> and autocorrelation function <span class="math-container">$R(t)$</span>. If <span class="math-container">$ R(t) \in L^1(0,\infty ) $</span>
then
<span class="math-container">$$ \lim_{T \to \infty} \Bbb E \  \bigg| { 1\over T} \int_{0}^{T} (X(s)-\mu) ds \ \bigg|^2=0$$</span></p>
<p>This result is intuitive: For <span class="math-container">$R(t)$</span> to be in <span class="math-container">$L^1(0,\infty)$</span>, <span class="math-container">$R(t)$</span> needs to fall off quickly. As a result, the process 'loses' memory quickly and forgets its initial value and keeps starting all over again and again. As a result, the process is ergodic.</p>
<p>(2) For a homogeneous Markov chain define operators <span class="math-container">$P_t$</span> via
<span class="math-container">$$ P_t f(x) = \Bbb E \left( f(X_t) \vert X_0=x \right) $$</span>
for suitable real-valued functions <span class="math-container">$f$</span> acting in the state-space of the Markov-chain.
Then <span class="math-container">$X_t$</span> is ergodic if
<span class="math-container">$$ P_t g=g$$</span> only admits constant solutions.In other words, the dimension of the eigenspace with eigenvalue 1 is one-dimensional.</p>
<p>Also, this result is intuitive since the dimension of this eigenstate corresponds to the number of clusters in the state-space: Whereas the dynamics of a particular path take place in a specific cluster only, it cannot generally agree with the expectation value since it averages over all clusters. Hence ergodicity implies one cluster only.(What about if one played devil's advocate and for some reason, the averages in all clusters are the same?)</p>
<p>What puzzles me is whether the two conditions are related and if so how.</p>
<p>PS: I posted this question also under Markov-chains/Dynamic-Systems but realize the question may be better off posted here</p>

CC BY-SA 4.0
2022-08-20T08:59:00.140Z
Something needs checking. Suppose that $X_t$ is a discrete-time process for simplicity and that $\mu=0$. Then $\int_t^{\infty} (X(t)-\mu)dt$ is simply $\sum_t^{\infty} X_t$, so the first displayed equation is not satisfied even for iid $X_t$ because $E[T^{-1} | \sum_t^{\infty} X_t |^2 ] = T^{-1} E(| \sum_t^{\infty} X_t |^2 ]) = T^{-1} \infty = \infty$ for all $T$. If the upper bound is changed to $T$, the condition is still not satisfied because $E[T^{-1} | \sum_t^T X_t |^2]$ is the variance of $X_t$ if $X_t$ is iid and $\mu=0$. The displayed condition is not even satisfied by iid processes.
chan1142
CC BY-SA 4.0
2022-08-22T07:16:08.827Z
Sorry but is it integral up to $T$ or up to $\infty$ in the edited formula?
chan1142
<econometrics><applied-econometrics>
Is autocorrelation and the number of clusters related in an ergodic process
user3072048
# Is autocorrelation and the number of clusters related in an ergodic process

(Asked by: username_0 on 2022-08-19)

I' ve come across two notions for the condition of (mean-) ergodicity and wonder whether and how they relate.

(1) Let $ {X}\_{t \ge 0} $ be a second order stationary process with (constant) mean $\mu$ and autocorrelation function $R(t)$. If $ R(t) \in L^1(0,\infty ) $
then
$$ \lim\_{T \to \infty} \Bbb E \ \bigg| { 1\over T} \int\_{0}^{T} (X(s)-\mu) ds \ \bigg|^2=0$$

This result is intuitive: For $R(t)$ to be in $L^1(0,\infty)$, $R(t)$ needs to fall off quickly. As a result, the process 'loses' memory quickly and forgets its initial value and keeps starting all over again and again. As a result, the process is ergodic.

(2) For a homogeneous Markov chain define operators $P\_t$ via
$$ P\_t f(x) = \Bbb E \left( f(X\_t) \vert X\_0=x \right) $$
for suitable real-valued functions $f$ acting in the state-space of the Markov-chain.
Then $X\_t$ is ergodic if
$$ P\_t g=g$$ only admits constant solutions.In other words, the dimension of the eigenspace with eigenvalue 1 is one-dimensional.

Also, this result is intuitive since the dimension of this eigenstate corresponds to the number of clusters in the state-space: Whereas the dynamics of a particular path take place in a specific cluster only, it cannot generally agree with the expectation value since it averages over all clusters. Hence ergodicity implies one cluster only.(What about if one played devil's advocate and for some reason, the averages in all clusters are the same?)

What puzzles me is whether the two conditions are related and if so how.

PS: I posted this question also under Markov-chains/Dynamic-Systems but realize the question may be better off posted here

**Tags:** econometrics, applied-econometrics
> **Comments:**
> **username_1**: Something needs checking. Suppose that $X_t$ is a discrete-time process for simplicity and that $\mu=0$. Then $\int_t^{\infty} (X(t)-\mu)dt$ is simply $\sum_t^{\infty} X_t$, so the first displayed equation is not satisfied even for iid $X_t$ because $E[T^{-1} | \sum_t^{\infty} X_t |^2 ] = T^{-1} E(| \sum_t^{\infty} X_t |^2 ]) = T^{-1} \infty = \infty$ for all $T$. If the upper bound is changed to $T$, the condition is still not satisfied because $E[T^{-1} | \sum_t^T X_t |^2]$ is the variance of $X_t$ if $X_t$ is iid and $\mu=0$. The displayed condition is not even satisfied by iid processes.  
> **username_1**: Sorry but is it integral up to $T$ or up to $\infty$ in the edited formula?  

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## 0 answers

