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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20 views

How to calculate average for financial random walk?

Let's look at a simple financial 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, ...
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367 views

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,...
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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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43 views

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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59 views

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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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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46 views

Stationary Process Ergodicity

Can you give me an example of a stationary nonergodic stochastic process that is time continuous?
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62 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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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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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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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 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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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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128 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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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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404 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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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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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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193 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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256 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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257 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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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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357 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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65 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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167 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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535 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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101 views

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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495 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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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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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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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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129 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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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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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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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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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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429 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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896 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 ...