# Questions tagged [ergodic]

An ergodic dynamic system or stochastic process is one in which time averages agree with averages over the state space of the process.

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### Why Asymptotic Equipartition Property theorem proofs assume the source is memoryless?

I do not understand the assumption 𝑋1,𝑋2,⋯ are i.i.d. ~p(x) in the AEP proofs I have seen. I have read some different sources for understanding the Asymptotic Equipartition Property. Using Cover &...
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### Gibbs sampler with adaptive linear transformation

It is a well known fact that linear transformations can dramatically improve the performances of a Gibbs sampler when a ridge-like joint likelihood function occurs. Can I make an algorithm that ...
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### Sampling a proposed value with a limited range target when running MCMC [duplicate]

I want to do an MCMC algorithm and need to sample a proposed value from a proposed distribution. In the Metropolis algorithm, people usually use a normal distribution as proposal. But if the prior ...
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### Are all ergodic random processes (at least wide sense) stationary?

If not, please provide a simple example of a non-stationary process that is ergodic (in mean and covariance).
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### Finding the distribution of a quantity using ensemble

In physics, we have the concept of ensemble average, which we use a lot in statistical physics. For example, ergodic property states that the time average of a statistical quantity is equal to its ...
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### Minimizing the rate of geometric ergodicity of a Metropolis-Hastings kernel

Let $\kappa$ denote the transition kernel of the symmetric Metropolis-Hastings algorithm with proposal kernel $Q$ and target distribution $\mu$. What would be a general guideline if we're aiming to ...
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### Is the Metropolis-Hastings kernel always aperiodic, irreducible and geometrically ergodic?

Let $(E,\mathcal E,\lambda)$ be a measure space, $Q$ be a Markov kernel on $(E,\mathcal E)$ with $$Q(x,B)=\int_B\lambda({\rm d}y)q(x,y)\;\;\;\text{for all }(x,B)\in E\times\mathcal E$$ for some ...
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### Does a Metropolis-Hastings chain always obey a central limit theorem?

Let $(E,\mathcal E,\lambda)$ be a measure space, $Q$ be a Markov kernel on $(E,\mathcal E)$ with $$Q(x,B)=\int_B\lambda({\rm d}y)q(x,y)\;\;\;\text{for all }(x,B)\in E\times\mathcal E$$ for some ...
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### Determine whether a time series is ergodic

How to determine if this time series is ergodic? According to the ergodic theorem the mean must converge to the expected value starting point. How do I determine that? Hoverever I have plotted the ...
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### Markov chain which is also a projection

Let $P$ be the transition matrix of a Markov chain, and assume that $P^2=P$. One immediate conclusion is that $P=P^\infty$. Furthermore, assume that there is a state $i$ such as each state $j$ (...
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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 ...
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### Within-chain and between-chain average difference in MCMC simulation

I have done 400 repetitions of a particular MCMC simulation (Metropolis–Hastings algorithm) to get a quantity of interest $N$. The simulation reaches its steady-state after ~$10^5$ iterations. The ...
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### Calculate of Lyapunov Exponents of a sequence of random matrices

Let $(\Omega,\mathcal{F},\mathbb{P}):=(M^{\mathbb{N}_{0}},\mathcal{M}^{\mathbb{N}_{0}},\mathbb{P})$ be a probability space where $M=\left\{0,1,2,3,4\right\}$, $\mathcal{M}^{\mathbb{N}_{0}}$ is product ...
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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 ...
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### Robert Casella Independence Sampler Result

In the second edition of the Robert & Casella book (Monte Carlo Statistical Methods), the authors have a result, Theorem 7.8, on the independent Metropolis-Hastings sampler: Letting $f$ be the ...
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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, $\sum_{n=0}^\infty p_{jj}^n = \infty$ ...
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### How do I measure a speedup?

I have two versions of the same program. I apply a number of performance tests to both and measure their response time, for instance https://gist.github.com/valtih1978/d2cc2fe96fbbe1987ada ...