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It is possible to show that elementwise convergence in probability implies $\text{plim}_{k \rightarrow \infty} (X_n^k) = (X_n)$ (i.e., convergence of the sequence). Thus, your conjecture effectively stipulates that the probability-limit of a sequence of strongly stationary sequences is also a strongly stationary sequence. Since convergence in probability ...

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In time series analysis, what they usually do is proving up to the second moment(including autocovariance). In case of Gaussian distribution, this proves the stationary. However, it doesn't guarantee the stationarity in general. Stationarity up to the second moment is so common and they even have a special name. They call it 'weak stationarity' (also, ...

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Here is my solution to "What is the simplest practical method to implement?" using python, specifically numpy, scipy and tick. One modification is that I set the exponential kernel such that alpha x beta x exp (-beta (t - ti)), to coincide with how tick defines exponential kernels: https://x-datainitiative.github.io/tick/modules/generated/tick....

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No that would be invalid: the prior has been learned from the data and so you are going to get an artificially narrow credible interval. One approach, if you think that the downstream parameters might be related to the upstream parameters, is to borrow strength across the two by using some kind of hierarchical model. For example if theta_u is the upstream ...

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If possible, you should switch to using the SARIMAX model, which has more features and will be better supported going forwards (the ARIMA model will be deprecated in the next release). The results object from a fitted SARIMAX model has a simulate method that allows you to simulate from the fitted process. If you want the simulation to start after the end of ...

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By definition, a stochastic process can be considered as a random variable that is evolving along time. Your "definition" is not nearly precise enough to allow you to consider the questions you're asking. Try looking up the actual standard definition and you'll find your questions have standard answers. A stochastic process is a function $$X(t, \... 0 This is an answer to the question Is this process ergodic? How could I demonstrate such thing? regarding the random process$$\{Y(t) = I(t)\cos(2\pi f_0t)−Q(t)\sin(2\pi f_0t)\} where $\{I(t)\}$ and $\{Q(t)\}$ are uncorrelated zero-mean stationary ergodic processes with identical autocorrelation function $R_I(z) = R_Q(z) = R(z)$. I will take ergodicity as ...

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It is neither. Probability theory and statistics are related, but distinct fields. Using the definitions from Encyclopædia Britannica: Probability theory, a branch of mathematics concerned with the analysis of random phenomena. [...] and Statistics, the science of collecting, analyzing, presenting, and interpreting data. [...] As you can see, ...

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If we know the probability distribution of data generation, we can calculate unconditional quantities directly without the help of observations. But we don't know the distribution and we have only one instantiation of time series in real life. They seek to find a way to calculate unconditional quantities from a time average over the time series observed. ...

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Hi: When you call the filter function in R, you're simulating the AR(2) model described at the top of your question. This statement of course assumes that you use the right function arguments so that you get the AR(2). There are parameters such as method, sides, circular, init etc and these need to be set correctly in order to simulate the AR(2). All the ...

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