# Questions tagged [stationary-processes]

For questions about strictly stationary or stationary in the wide sense sequences or processes. Questions about deterministic stationary processes (in the case of discrete dynamical systems) are welcome.

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### $\beta$-mixing in Asmptotically Stochasitic (Random) Process

This issue involves a very important concept, which is the $\beta$-mixing nature of stochastic processes. All the stochastic processes we discuss are time-positive and discrete. To strictly adhere to ...
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### Can anyone help to solve this task ?In a multiple-choice test with m options, a student knows the correct answer with probability p,...?

"In a multiple-choice test with m options, a student knows the correct answer with a probability p, and in the absence of knowledge, chooses randomly one of the available options. What is the ...
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### Expression of the stationary distribution of a Markov Chain (PageRank)

I want to find the expression for the PageRank of a webpage defined as in the original paper of Sergey and Larry (The Anatomy of a Large-Scale Hypertextual Web Search Engine). Consider a directed ...
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### Are transition probabilities always absolutely continuous w.r.t. invariant measure?

Let $X_n$ be a Markov process with transition kernel $p$. A probability measure $\mu$ is called invariant (or stationary) if $$\int p(x,A)\,d\mu(x) = \mu(A)$$ for all measurable sets $A$. My ...
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### Finding the spectral measure of a of a weakly stationary process

For these exercices, I am asked to find the spectral measure of their processes if they are weakly stationary. However, I do not understand how to do so. https://i.sstatic.net/vUQIB.png For exemple, ...
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### Difference between time reversible and stationary distribution. "watching a movie backwards"

A time homogeneous Markov process on Ω with semi-group $P_t$ is said to stationary w.r.t a distribution $π$ if $$∫_{Ω}P_tf(x)dπ(x)=∫_{Ω}f(x)dπ(x), \text{for f bounded measurable}.$$ and reversible ...
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### Expectation of product of 3 samples of WSS Gaussian process

Suppose $X_n$ is a WSS Gaussian stochastic process with(every subsample of the process is jointly gaussian): $E[X_n]=\mu$ for all n $R_{xx}[l]$ is the autocorrelation function I wish to calculate : ...
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### Can the Wiener-Khinchin theorem be correctly applied to a periodic sound signal (such as a sine wave)?

The theorem speaks about a wide-sense stationary random process. Is, for example, a sine wave with a period 1/400 s considered a WSS (or, in general, a periodic sound signal with multiple frequency ...
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### Power Spectrum Definitions

Consider a stationary random process $x(n)$ with $n = 0, \pm 1, \pm 2,..$ The Autocorrelation function is defined as: $$R_{xx}(m) := \mathbb{E}[x^*(n)x(n+m)]$$ where the * denotes conjugation. I want ...
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### Weak convergence of $\sum_{k=0}^n \lambda^k X_{n-k}$ if $X$ is a stationary process
Suppose that $(X_k)_{k \in \mathbb{N}}$ is a stationary Markov chain with state space $\mathbb{R}$ and $0<\lambda<1$. Can we say something about the weak convergence of the sequence \$Y_n = \sum_{...