17
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Ergodicity explained in layman terms
Here's the simplest way I can think of: if you watch a stochastic process long enough you're going to see every possible outcome. Not only that, but also you can obtain the probabilities of such ...
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10
votes
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Why is an unbiased random walk non-ergodic?
That Wikipedia article writes,
The process $X(t)$ is said to be mean-ergodic or mean-square ergodic in the first moment if the time average estimate $${\hat {\mu }}_{X}={\frac {1}{T}}\int _{0}^{T}X(...
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7
votes
Magnitude of non-ergodicity effect on the individual's risk of bankruptcy
Ole Peter's presented the same material here with minimal changes, and I tried again to replicate the code, forgetting that I had posted this question until I searched for references to Ole Peter's ...
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6
votes
How do you check ergodicity of a stochastic processes from its sample path(s)?
A signal is ergodic if the time average is equal to its ensemble average. If all you have is one realization of the ensemble, then how can you compute the ensemble average? You can't. Therefore you ...
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5
votes
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What does the distribution of samples from an MCMC method converge to without repeated samples?
We addressed this problem in our 2011 vanilla Rao-Blackwellisation paper. The limiting distribution of the unique simulations in the Metropolis-Hastings sequence is associated with the density
$$\...
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5
votes
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Metropolis Hastings with estimated posterior
I do not understand the need for this construction if you can
simulate from $P(\phi|x)$
simulate from $P(\theta|\phi)$
since neither an approximation nor an MCMC implementation is then required.
...
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4
votes
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Markov Chains: Periodicity and Ergodicity
As understood in our book, ergodicity means convergence to the stationary distribution of the Markov chain irrespective of the initial condition or distribution. Therefore, if a Markov chain is ...
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4
votes
Ergodicity explained in layman terms
To quote Wiki with a little formatting of my own:
[...] a stochastic process is said to be ergodic if its statistical properties can be deduced from a single, sufficiently long, random sample of ...
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4
votes
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Why is ergodicity not a requirement for ARIMA models besides stationarity?
A bit technical maybe, but stationary ARMA processes are by construction mean-ergodic (as the other answer correctly pointed out, a previous version of my answer did not spell that out clearly and ...
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4
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Stationarity and Ergodicity - links
Ergodicity is a property defined for strictly stationary processes, i.e. an ergodic process is by definition strictly stationary.
Note The property being shown by the answer in Why is ergodicity not a ...
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3
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Why is ergodicity not a requirement for ARIMA models besides stationarity?
Ergodicity and mean-ergodicity are not the same properties.
Ergodicity is a much stronger property than mean-ergodicity (mean-ergodicity just means an $L^2$-LLN holds). There are easy examples of ARMA ...
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3
votes
Accepted
Ergodic theorem
If the state $j$ is recurrent for the Markov chain $(X_t)$, $$\sum_{n=0}^\infty p_{jj}^n = \infty$$is equivalent to [in the sense that the lhs is the same] $$\sum_{n=0}^\infty \mathbb{P} (X_n=j|X_0=j) ...
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3
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Why is $\frac1n\sum_{i=0}^{n-1}\sum_{j=0}^i1_{\{\:X_i\:=\:Y_j\:\}}f(Y_j)$ an equivalent representation for the usual Metropolis-Hastings estimator?
As you say, it is sufficient to show that, with probability $1$, all proposed points $Y_t$ are distinct. Note that the fact that the $X_t$ come from a Markov chain is inessential to showing that the $...
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Ergodicity explained in layman terms
From Bishop, a Markov chain is ergodic when you can run it starting from any initial distribution and end up converging to its invariant distribution (steady state, or equilibrium).
A sufficient ...
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3
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How are ergodicity and "weak dependence" related?
I had the same question, and found these lecture notes. Page 8 states that a mixing process is ergodic (called Theorem 7) and that a mixing process is also called weakly dependent.
In other words, a ...
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How are ergodicity and "weak dependence" related?
The concepts are not interchangeable. Ergodicity deals with studying the systems where different realizations of the process are not available. For instance, in coin toss we could reasonably argue ...
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3
votes
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Wide-Sense Stationary but not ergodic
Let's leave aside the concept of ergodicity itself, which is rather deep, abstract and difficult, and use its consequences: when a stochastic process $\{X_i\}$ is ergodic, the "time" average (the ...
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3
votes
Accepted
ergodic theory for markov processes
Say your state space is $\Omega$ and your process is $X_{t}$. Consider now a new state space - $\Omega \times \Omega$. Then $Y_{y} := (X_{t-1}, X_{t})$ is a Markov process on $\Omega \times \Omega$. ...
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3
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Difference between stationarity and ergodicity
A very simple example of a series which is stationary but not ergodic is the following:
$$ X_t = \sin(2\pi t + \Omega), \quad t=0,2,3,\dotsc $$
where $\Omega \sim \mathcal{Unif}(0, 2\pi)$. Observe ...
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3
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Difference between stationarity and ergodicity
Adapted from an answer of mine on dsp.SE, with editing to suit the sensitivities of stats.SE readers....
A random process is a collection of random variables, one for each time instant under ...
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3
votes
Accepted
Stochastic modelling, distribution and ergodicity of a particular time series with a given finite history
Your question misunderstands how conditioning works in measure-theoretic probability
I think you are letting the complexity of the notation for a stochastic process get in the way of intuitive ...
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2
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Classification of states in Markov Chain
Let the state space of the Markov Chain be $S=\{1,2,3,4,5,6\}$. Now draw the state transition diagram.
(a). From the figure, we observe that $\{4\}$, and $\{6\}$ form non-closed communicating classes....
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Examples of ergodic process
Look at a video of the behavior of, say, an adult male ant for 20 minutes. Would you have noticed if after 10 minutes, the ant was replaced by another adult male ant?
If you wouldn't notice because ...
2
votes
Why is $\frac1n\sum_{i=0}^{n-1}\sum_{j=0}^i1_{\{\:X_i\:=\:Y_j\:\}}f(Y_j)$ an equivalent representation for the usual Metropolis-Hastings estimator?
The proposals are coming from a density, so they should all be different with probability one.
It might make it clearer if you write out a sample path from the algorithm. Think of the chain on the ...
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2
votes
Accepted
Law of Large Numbers for Covariance Stationary Processes... Difference and Relationship between LLN and Ergodicity
I'm dealing right now with this kind of topic for my last physics master degree exam, so I'll try to give what I think is the best way to tackle the question.
Ergodicity in general is, as you stated, ...
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Stationarity and ergodicity of a process conditional on a finite trajectory
Your answer here is mostly technically correct (though it is of course possible for some time-series that you might get observed values $y_1= \cdots = y_t = \mathbb{E}(X_k)$ in some other applications)...
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vote
Proof of Autocorrelation function property
A very hand-waiving but still could be powerful way of visualizing this property could be to list that:
$$ \mu_x(t)=E[x(t) ]=\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}x(t) dt=\widehat{\mu_x(t) }$$
for ...
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vote
Stationary Process Ergodicity
Let $X_0$ have a Bernoulli$(p)$ distribution, $0\lt p\lt 1,$ and define $X_t=X_0$ for all $t.$
It is stationary because all finite-dimensional joint distributions of $(X_{t_1}, \ldots, X_{t_k})$ are ...
- 323k
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vote
How do I create an iid Rademacher sequence?
This is just a change of variable calculation. Use the CDF method. The CDF of a Rademacher RV is just $0$ if $x < -1$, $0.5$ if $-1 < x < 1$ and $1$ if $x > 1$. You can show $2\omega - [2\...
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vote
Ergodicity explained in layman terms
This is an answer to the question
Is this process ergodic? How could I demonstrate such thing?
regarding the random process $$\{Y(t) = I(t)\cos(2\pi f_0t)−Q(t)\sin(2\pi f_0t)\}$$
where $\{I(t)\}$ ...
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Only top scored, non community-wiki answers of a minimum length are eligible