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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.

23 questions with no upvoted or accepted answers
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128 views

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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28 views

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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66 views

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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25 views

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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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 ...
2
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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 $j$ (...
2
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0answers
34 views

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 ...
2
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0answers
77 views

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 ...
2
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0answers
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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35 views

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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28 views

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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60 views

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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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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48 views

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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22 views
+50

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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1answer
48 views

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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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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26 views

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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1answer
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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24 views

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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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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98 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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40 views

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 $...