An ergodic dynamic system or stochastic process is one in which time averages agree with averages over the state space of the process.
Wikipedia has an article https://en.wikipedia.org/wiki/Ergodicity with further references.
Questions tagged [ergodic]
68 questions
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votes
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How do you check ergodicity of a stochastic processes from its sample path(s)?
How do you check ergodicity of a wide-sense stationary stochastic processes from its
sample path(s)?
Can we check ergodicity from a single sample path? Or do we need multiple sample paths?
One motivation of checking ergodicity is in time…
Tim
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Ergodicity explained in layman terms
I've been told that Ergodicity gives us a practical vision of processes WSS (Wide-sense stationary) and a bunch of integrals. For me, it is not enough to fully understand it.
Could someone explain me Ergodicity in a simple way?
EDIT:
Thank you all…
WhiteGlove
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Why is an unbiased random walk non-ergodic?
Wikipedia says "An unbiased random walk is non-ergodic."
Let's look at a simple random walk. It's defined as: take independent random variables $Z_{1},Z_{2}$, where each variable is either $1$ or $−1,$ with a 50% probability for either value, and…
Alex Craft
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11
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Is there a Monte Carlo/MCMC sampler implemented which can deal with isolated local maxima of posterior distribution?
I'm currently using a bayesian approach to estimate parameters for a model consisting of several ODEs. As I have 15 parameters to estimate, my sampling space is 15-dimensional and my searched for posterior distribution seems to have many local…
akraf
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AcF and Stationarity
Very often in time series literature, it is remarked that if a series is non-stationary the AcF will decrease to zero very slowly while the opposite occurs for a stationary series.
What's the basis for this "rule of thumb"? I know that for a…
gamerx
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Why is ergodicity not a requirement for ARIMA models besides stationarity?
I frequently read that ARIMA models must be fitted on stationary data. But stationarity does not ensure ergodicity, which I understand is necessary to deduce population parameters from a single time series sample. Why is ergodicity not a requirement…
JTicker
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When is a ARMA(p,q) process ergodic?
We know that a ARMA(p,q) process is weakly stationary, iff there is no root of the characteristic polynomial of its AR part lying on the unit circle.
But what is the necessary and sufficient condition for a ARMA(p,q) process to be ergodic? Any book…
Tim
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6
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ergodic theory for markov processes
For an ergodic Markov Chain
$$
\frac{1}{N}\sum_{i=1}^n f(X_i) \rightarrow E_\pi[f]
$$
where $\pi$ is the invariant distribution. I am also dealing with a Markovian process (a state space model to be specific) and I have a quantity like the…
jkt
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5
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What does the distribution of samples from an MCMC method converge to without repeated samples?
Suppose I have an absolutely continuous distribution with density $f(x)$ and I use an mcmc sampler which has accept/reject step to sample from this distribution. In the final samples, there are some singular points (i.e., samples on top of each…
KiaSh
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How are ergodicity and "weak dependence" related?
I understand that weak dependence is a broad concept, the definition I am referring to is the one Wooldridge (2013) uses as an assumption that has to be fulfilled (amongst other assumptions) so that the estimators in a time series linear regression…
shenflow
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Derivation of sample autocovariance
The autocovariance is defined as
$$\gamma(t,s) = Cov(X_{t}, X_{s})=E[(X_{t}-\mu_{t})(X_{s}-\mu_{s})]$$
When we have a stationary process the only thing that matters is the lag between the variables:
$$\gamma_{k} = Cov(X_{t},…
Robert Smith
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0 answers
Stationary Distribution of Multiplicative Autoregressive Model
I know for the additive autoregressive model the stationary distribution of $\{X_t\}$ can be found, if it exists, in the following way:
\begin{align}
X_t &= \alpha X_{t-1} + \epsilon_t\\
\Rightarrow X_t &= (1-\alpha B)^{-1} \epsilon_t \\
&=…
Shanks
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Examples of ergodic process
An ergodic process is a process in which the structures of inter-individual variation and intra-individual variation are asymptotically equivalent (Molenaar, 2004).
In other words:
A process is non-ergodic in case results of analysis of…
JetLag
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4
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Metropolis Hastings with estimated posterior
I am interested in samples of $\theta$ from the posterior distribution
$$
P(\theta|x) = \int d\phi P(\theta|\phi)P(\phi|x)
$$
where $x$ are data and $\phi$ are nuisance parameters. In principle, I can use a Metropolis Hastings sampler to sample from…
Till Hoffmann
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Ergodic theorem
We know that,
If $p_{jj}$ is the transition probability of staying in state j at n-th step and j at (n-1)-th step then the state j is said to be recurrent if,
$\sum_{n=0}^\infty p_{jj}^n = \infty$
and transient if,
$\sum_{n=0}^\infty p_{jj}^n \lt…
Rhafi Sheikh
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