# 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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### 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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1 vote
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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 ...
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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 ...
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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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