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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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Ergodicity-definition for general statistic

I'm struggling with the definition of ergodicity within time series. Consider a time series denoted as $X = (X_i)_{i\in\mathbb{Z}}$, where each $X_i$ represents a random vector defined on the same ...
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Invariant event defined in terms of stationary stochastic sequence

In "Almost Sure Convergence" by Stout, there is indicated that the concept of invariant event (and further, the concept of ergodicity) can be defined in terms of given stationary stochastic ...
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Is there any reference about Ergodic Theorems applicable to stochastic processes with strong dependence?

Consider the stochastic process $(X_{n})_{n\in\mathbb{N}} = (A^{+}_{n},A^{-}_{n})_{n\in\mathbb{N}}$ defined over the same probability space $(\Omega,\mathcal{B},\mathbb{P})$ such that the occurrence ...
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What is the correct definition of the Cesàro summation of autocovariances?

I'm a little confused regarding a mathematical definition (Cesàro summation) and its application to stationary time series. First, consider the definition given by Wikipedia, adapting to autovariances....
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Sufficient conditions for second-moment ergodicity

Let $(Y_t)_{t \in Z}$ be a covariance-stationary stochastic process. According to Hamilton (page 46-47), we say that the process is Ergodic for the mean if $$\overline{Y}\equiv \frac{1}{T}\sum_{t=1}^...
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What do I have to show still to prove that $x_t$ is stationary and ergodic?

Let $x_t=\varphi x_{t−1}+\varepsilon_t$ be the model with errors being white noise. If model is correctly specified and $|\varphi|<1$, why is $x_t$ not stationary and ergodic? Why cannot I use ...
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Is a stable VAR model geometrically ergodic

If we have a VAR process: $\begin{align} \mathbf{y}_t = A_1\mathbf{y}_{t-1} + \dots + A_d\mathbf{y}_{t-d} + \boldsymbol{\epsilon}_t, \quad t \in \mathbb{Z} \end{align}$ With the stability condition ...
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Can I add more steps to a Gibbs sampler without hurting the ergodicity of the chain?

I have a Gibbs sampler that updates a system of $n$ variables $(x_1,\ldots,x_n)$ by each of the full conditional distributions. Let's say that I add a $n+1^{th}$ step: I also update $v^T(x_1,\ldots,...
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Uniform ergodicity of a Gibbs sampler

We consider a classical data set from Gelfand and Smith containing the information about ten nuclear power plant pump failures. We are interested in the failure intensity of each pump and we employ ...
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What properties does a mean-ergodic and non ergodic for second moments time series have?

The first and the most trivial example of a mean-ergodic stochastic process is the i.i.d. case: $$(u_t)_{t \in \mathbb Z} \sim N(0,1)$$ The first intuition we have of ergodicity is that the process $(...
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Which convergence to use when showing no-mean-ergodicity?

This is a basic econometrics question about ergodicity. According to Hamilton, we say that the stationary time series $(X_t)_{t \in \mathbb Z}$, , with mean $\mu$, is mean ergodic if the following ...
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How to compute the ergodic mean of a series?

In a paper, the authors say, "We can thus approximate welfare by computing the ergodic means of log consumption and log consumption growth from long simulations of the model.". They use the ...
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About ergodic stochastic process

$x_t=2.1+0.73x_{t-1}+ε_t$ $ε_t \sim iid(0,σ_ε^2)$ Given the stochastic process with deep Gaussian process above, is x_t an ergodic stochastic process? If possible, I would like to know the reason.
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Stationarity and ergodicity of a process conditional on a finite trajectory

Let us say we are interested in a single time series, e.g. the daily closing share price of Tesla. We can model it as a realization of a stochastic process $\{Y_t(\omega)\}$. It corresponds to a ...
Richard Hardy's user avatar
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Stochastic modelling, distribution and ergodicity of a particular time series with a given finite history

Let $\Omega$ be a sample space. A stochastic process $\{Y_t\}$ is a function of both time $t \in \{1, 2, 3, \ldots\}$ and outcome $\omega \in \Omega$. For any time $t$, $Y_t$ is a random variable (i....
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Difference between stationarity and ergodicity

I am currently studying time series. However, I have trouble understanding how ergodicity and stationarity differ. Could someone clarify the difference between the two concepts in simple terms and ...
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Ergodic theorem for Markov chains

I am reading Robert and Casella (2004) on Markov Chain Monte Carlo methods and, in particular, Section 6.7. This contains the ergodic theorem, which is stated as follows, where $S_n(f)$ denotes a ...
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Is ever a covariance stationary process, in the real world, not mean ergodic?

Mean ergodicity for a covariance stationary process is a property that assures that the sample time mean $\bar{Y}=\frac{1}{T} \sum_{t=1}^T Y_t$ converges as $T \rightarrow \infty$ to the ensemble ...
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Classification of time series according to linearity, ergodicity and stationarity

I'm new to the subject of Time Series and I have some difficulties understanding some cross definitions: Linear and non-linear time series Ergodic and non-Egordic time series Stationary and non-...
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What are the requirements for a Markov chain to have a stationary distribution?

I read about Markov chains in quite a lot of different resources. However, I can't seem to find a consistent definition of what the requirements are for a Markov chain to have a stationary ...
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Does the convergence rate never increase of a Stationary Ergodic Random Processes under sub-sampling?

Summarize the problem Given A Stationary Random Processes (strict sense) $X_i$ I define two Stationary Ergodic Random Processes by $$ \bar{X}_n = \frac{1}{n} \sum_{i=0}^{n-1} X_i \ \ \text{and} \ \ \...
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HAC variance to construct standard errors

I am facing some difficulties understanding this question. It hasn't been long since I started with econometrics, so I'm new to all of this. Suppose we have a function $$E[c_t|y_t,c_{t-1},y_{t-1},c_{t-...
Maybeline Lee's user avatar
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Proof of Autocorrelation function property

I am trying to intuitively make sense of a specific property of the autocorrelation function: $$ \lim_{\vert\tau\vert \rightarrow \infty} R_{XX}(\tau) = \bar{X}^2 $$ where $\bar{X} = \mathbb{E}[X(t)]$ ...
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Does Excess Kurtosis Signal Non-Ergodicity?

I have been reading a lot about ergodicity and the main principle behind it seems pretty simple actually. Based on my understanding, something is ergodic if the time average and the ensemble average ...
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Is this adaptive MCMC approach ergodic?

I am aware that adaptive MCMC proposals can be dangerous, as they can affect the ergodicity of the chain and hence potentially invalidate the results obtained. Andrieu & Thoms (2008) (DOI 10.1007/...
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Stationarity and Ergodicity - links

In time series analysis stationarity and ergodicity have different definitions and meanings: https://en.wikipedia.org/wiki/Stationary_process https://en.wikipedia.org/wiki/Ergodic_process Essentially ...
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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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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,...
Alex Craft's user avatar
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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 ...
yu zhang's user avatar
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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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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 ...
UmaN's user avatar
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Stationary Process Ergodicity

Can you give me an example of a stationary nonergodic stochastic process that is time continuous?
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Are explosive AR(MA) processes stationary?

According to Theorem 8.8 in Time Series by A.W. van der Vaart, an ARMA process $$ \phi (L)X_t=\theta(L)\epsilon_t $$ has a unique stationary solution $X_t=\psi(L)\epsilon_t$ with $\psi=\theta/\phi$ ...
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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 ...
Chaos's user avatar
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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 ...
ColorStatistics's user avatar
5 votes
1 answer
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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\...
Antoni Parellada's user avatar
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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 (Wide-sense stationary) and a bunch of integrals. For me, it is not enough to fully understand it. Could someone explain me ...
WhiteGlove's user avatar
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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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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 ...
M. Hansen's user avatar
6 votes
2 answers
978 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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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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