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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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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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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?

At the beginning of section 2 of the paper A Vanilla Rao-Blackwellization of Metropolis-Hastings Algorithms, the usual Metorpolis-Hastings estimator of $\int f$ given by the ergodic average $\frac1n\...
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39 views

Is Autocorrelation of Posterior Samples always a problem in MCMC

I am experimenting with MCMC methods and have implemented a basic Metropolis-Hastings algorithm. One potential issue with this is that MH posterior samples are autocorrelated. I could verify that ...
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1answer
229 views

Law of Large Numbers for Covariance Stationary Processes… Difference and Relationship between LLN and Ergodicity

We have a covariance stationary time series. We must assume that the time series was produced by an ergodic process if we are to make the bridge between the realization of the time series that we ...
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44 views

Stationary Process Ergodicity

Can you give me an example of a stationary nonergodic stochastic process that is time continuous?
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329 views

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 ...
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40 views

What is Ergodic Variance

I am curious as to the definition of ergodic variance in relation to an estimate of some parameter. It was mentioned to me by a teacher although I have not been able to find any references to it.
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46 views

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 ...
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Do measurable maps preserve stationary ergodicity?

In a recent effort to establish stationary ergodicity for a certain stochastic process, I just happened to come across a statement, which I find to be little bit confounding. Given two measurable ...
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Do we need ergodic-stationarity of the response variable in OLS spline regression?

I was wondering if we need the response variable to be ergodic stationarity when estimating an OLS spline regression. My intuition tells me that it's not needed but I would like to have a confirmation ...
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22 views

Prior/degree of belief/degree of lack-of-information/algorithms/complexity

For a long time I had a bit of difficulty understanding what "degree of belief" means. Recently I had some thoughts about it and I wonder if they make any sense, or is there some literature about ...
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Stationary and ergodic r.v: relation between error and independent variable

In time series often hold the condition that a r.v. is stationary and ergodic, allowing the application of the law of large number. If in a model as: Y= a + bX + u ...
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114 views

Resource request : How to prove the output of a process is random variables?

I am reading through articles which present the spectral properties of chaotic systems such that they can be candidates for generating pseudo random binary sequences. One such article, is http://...
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348 views

Is it necessary to check ergodicity in estimation of autocorrelation function?

Given a sample path from a process supposed to be stationary, I saw the sample autocorrelation function of the sample path is used to estimate the autocorrelation function of the process. But this ...
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When are time averages equal to statistical averages?

I have a data set which comprises N measurements. Each measurement is an 8 dimensional vector representing 8 voltages measured from a machine. I want to compute the covariance matrix of this data. ...
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Magnitude of non-ergodicity effect on the individual's risk of bankruptcy

Dr. Ole Peters presents the concept of (non-)ergodicity with the following gambling example: You're given $\$100$ to play a game where you toss a coin once a minute. If it comes up heads, you win $50\...
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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 ...
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1answer
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How do I create an iid Rademacher sequence?

The lecture notes say: Let $(\Omega,\mathcal{A},P) = ((0,1],\mathcal{B}((0,1]),\lambda)$ where $\lambda$ is the Lebesgue measure on the unit interval. Define $X(\omega) = 1$ for $\omega > 1/2$ ...
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969 views

Ergodicity explained in layman terms

I've been told that Ergodicity gives us a practical vision of processes WSS (Wise-sense stationary) and a bunch of integrals. For me, it is not enough to fully understand it. Could someone explain me ...
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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\\ \...
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1answer
111 views

Is strict stationarity a sufficient condition for ergodicity?

For a given time series, is strict stationarity a sufficient condition for ergodicity? I am wondering if it isn't also sufficient for a time series to be weakly stationary because then the mean is a ...
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302 views

Wide-Sense Stationary but not ergodic

This is based on example 2.2 from Machine Learning: A Bayesian and Optimization Perspective by Theodoridis. Please note that I'm not at all familiar with ergodic theory and I'm reading this with ...
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Is the matrix $\hat X_k\hat X^T_k$ ergodic?

Let $X_k = A_{k-1}X_{k-1} + \omega_{k-1}$, and $Y_k = H_{k} X_k +\eta_k$, where $\omega_k$ and $\eta_k$ are i.i.d. Gaussian random variables. We now can get unbiased estimate of the random variable $...
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59 views

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-...
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161 views

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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204 views

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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206 views

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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85 views

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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1answer
813 views

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 ...
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1answer
328 views

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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1answer
63 views

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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412 views

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 ...
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1answer
154 views

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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1answer
454 views

Markov Chains: Periodicity and Ergodicity

I found the dynamical system definition of ergodicity to be very intuitive: $T$, a measure preserving transformation wrt $(\Omega,\mu)$ is ergodic wrt $\mu$ if for all $A \subset \Omega$, $T^{-1}A = ...
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Transformations on sets of measure zero in probability space

Given a probability space $(X, \Sigma,\mu, T)$, where $\mu$ is a probability measure, $\Sigma$ is a sigma algebra on $X$, and $T$ is a transformation, then a 'measure preserving transformation' is one ...
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A question about ergodic processes - 'global' vs 'local'

I have a question regarding the ergodicity of a transformation. Given a probability space, $(X,\Sigma,\mu,T_{N})$ where, $({T_{N})}_{n\geqslant 0}$ And T is some ordered transformation ...
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2k views

Classification of states in Markov Chain

Question Consider the following transition matrix: ...
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1answer
71 views

Problem involving Ergodic theorem and Markov Chain

With regards to the question in the above picture and the markov chain drawn in the question, my query is whether is it possible to conclude from Ergodic theorem that this Markov chain has an ...
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1answer
137 views

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 ...
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1answer
918 views

How to detect if Ergodicity, Stationarity and Martingale. dif. sequence?

I'm not sure, but I think I've read somewhere that because the Classical Linear Regression model assumes to have a random sample, when researchers they might not be in presence of a sample with that ...
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1answer
122 views

Introduction to Markov Process (Part 2): Dynamical system and Markov relationship?

In my previous post asked here Introduction to Markov process: How to prove that a process is Markov? Part 1, -- a process is Markovian if it follows the memoryless property. Consider, a dynamical ...
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1answer
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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 ...
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How do I test a (mathematical) system of equations for ergodicity?

I have a set of equations, which I can plot in for example Matlab. I would like to test for ergodicity: how can I do this? Similar questions have been asked here and here, but it is not very clear ...
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1answer
141 views

Ergodicity of 2 independent ergodic random processes

I'm wondering if $\{X_i\}$ and $\{Y_i\}$ are 2 independent processes that are ergodic, then would $\{(X_i,Y_i)\}$ be ergodic? I believe it is the case under the additional assumption that the two ...
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How long does it take two identical hidden Markov models run on same observations to forget their initial distributions (if ever)?

Let $H_1$ and $H_2$ be two instances of a finite Hidden Markov Model (HMM) $H$. That is, $H_1$ and $H_2$ have identical state spaces $Q$ as well as identical transition $A$ and emission probabilities $...
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393 views

What does non-ergodicity mean for Bayesian statistics?

As said in the title, what would non-ergodicity mean for Bayesian satistics, and if the process being investigated is non-ergodic, how would Bayesian methods tackle this process - would it be ...
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1answer
127 views

Metropolis Ergodicity

I have encountered one last problem with regarding to the Metropolis-Hastings algorithm. I know that ergodicity is needed in the algorithm to imply convergence to a unique stationary distribution. But ...