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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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$\mu(x)\mathbb P_x(X_n=y,n<\tau)=\mu(y)\mathbb P_y(X_n=x,n<\tau)$

Suppose that we are given a discrete time Markov chain with reversible measure $\mu$. By definition, $$\mu(x)\mathbb P_x(X_1=y)=\mu(y)\mathbb P_y(X_1=x)$$ and by induction we even get$^1$ $$\forall n:\...
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Convergence of subsequences of a stationary stochastic process

Let $\{X_n\}_{n=1}^\infty$ be a real-valued stationary stochastic process, and let $\{W_n\}_{n=1}^\infty$ be a binary-valued stochastic process, where $W_n \in \{0, 1\}$. We call $W_n$ the event ...
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The Box-Jenkins methodology provides two principles to select among equivalent models.1. Parsimony,2. Invertibility

In Invertibility -When choosing parameters in MA processes (either pure MA or ARMA), always select parameter values so that the MA coefficients are invertible. Because any MA(q) model has 2^q ...
THE FANTASTIC's user avatar
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2 answers
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Probability 2 earthquakes happen in a period of time.

The amount of earthquakes that happen at island X follows the Poisson process with mean 2 . Given that 2 earthquakes have happened in this year, find the probability both the earthquakes happen ...
user1259172's user avatar
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Regarding 1D Asymmetric Simple Exclusion Processes

I have been trying to decipher a paper on Asymmetric Simple Exclusion Processes in 1D by B. Derria: "An exactly soluble non-equilibrium system: The asymmetric simple exclusion process". ...
mathphyguy's user avatar
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Are ergodic continuous time processes (strictly) stationary in the limit?

If $X_t$ is a continuous time Markov process in a general state space $X$, say $X=\mathbb{R}^d$. Is it necessarily true that $X_t$ is stationary in the limit, as I believe that any ergodic process ...
Daan's user avatar
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Hidden Markov Model of a stationary Markov chain is a stationary process.

Let $X=(X_i)_{i \ge 1}$ be an irreducible Markov chain started in its stationary distribution, and $Y=(Y_i)_{i \ge 1}$ be such that $Y_i=\phi(X_i)$ for an arbitrary function $\phi$. Note that $X$ is a ...
hegash's user avatar
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Representation of stationary joint distribution by a fixed function of iid variables

Suppose $(X_t)_{t\in \mathbb{Z}}$ is a stationary real-valued sequence. Fix a positive integer $d$ and take $\mathbf{X}=(X_1,\ldots,X_d)$. The questions is, does there always exist a measurable ...
Uchiha's user avatar
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5 votes
2 answers
130 views

Ergodic series converge to the expectation?

Let $(X_i, Y_i)_{i\in\mathbb{N}}$ be a real-valued stochastic process. We say that $X$ is mean-ergodic, if $$\frac{1}{n}\sum_{i=1}^nX_i\to \mathbb{E}X_1$$ in probability as $n\to\infty$. Let $S_n:=\{i\...
Albert Paradek's user avatar
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Can a stationary process have "gaps"?

A process $\{X_t\}$ is said to be strictly stationary if $$ F_X\left(X_{t_1},...,X_{t_n}\right) = F_X\left(X_{t_{1}+\tau},...,X_{t_n+\tau}\right) $$ holds for all points in time $t_1,...,t_n\in \...
mto_19's user avatar
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Problem with understanding the expected value $E(y_t)$ in definition of a stationary time series.

I am trying to understand the conditions for time series $\{y_t\}$ to be stationary, i.e.: $E(y_t)=\mu$ is constant for all $t >0,$ $Var(y_t) = V$ is constant for all $t >0$ and $cov(y_t, y_s)$ ...
Brzoskwinia's user avatar
3 votes
1 answer
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Requirements for a Markov chain to converge to its stationary distribution.

I have seen in two places, different requirements for a Markov chain to converge to its stationary/invariant distribution: Irreducibility and aperiodicity. As mentioned here Irreducibility and ...
Dylan Dijk's user avatar
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Two-time correlation function from probability distribution

Given a continuous stochastic process $x_t$ defined by the following Langevin equation \begin{equation} d x_t = dB_t +F(x_t)dt \end{equation} where $dB_t$ is a Wiener increment and $F(x_t)$ is a ...
J.Agusti's user avatar
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2 votes
1 answer
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Top to random shuffling stationary distribution

Top to random shuffling is a method of shuffling a deck of N cards whereby the top card of the deck is removed and placed at random in the deck, and the procedure is repeated. I want to know the ...
abc's user avatar
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Why $(0.5,0.5)$ isn't stationary distribution for every 2 state ergodic Markov chain?

Assume M is 2 state- ergodic Markov chain - i.e it's irreducible and aperiodic, with Stochastic matrix $Q \in \mathbb{R}^2 $ . Then $Q(1,1) = (1,1)$. Hence $Q(\alpha,\alpha) = (\alpha,\alpha)$. Now, ...
Ron Abramovich's user avatar
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When do mixtures of ergodic Markov kernels remain ergodic?

Given two Markov kernels on the same space $\mathfrak X$ and relative to the same dominating measure, $K_0(\cdot,\cdot)$ and $K_1(\cdot,\cdot)$, both ergodic with respective stationary distribution ...
Xi'an ні війні's user avatar
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Decomposing stationary point process $Y$ as sum of two point processes $X,Z$. Can $X$ be chosen stationary as well?

Consider the following setup. We are given two $[0,1]$-marked point processes on $\mathbb R$ or $[0,\infty)$, denoted $X,Y$. Denote that the ground processes (i.e. the point processes ignoring marks) ...
Václav Mordvinov's user avatar
6 votes
1 answer
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Show that $X_{t}:=\alpha X_{t-1}+\epsilon_{t}$ is strictly stationary for $|\alpha|<1$ and $\epsilon_{t}$ i.i.d$~\sim N(0,\sigma^{2})$.

The title can be shortened to "prove that $AR(1)$ processes are strictly stationary when $|\alpha|<1$". This has been discussed many times on MSE and Cross Validated, but I found no ...
JacobsonRadical's user avatar
1 vote
1 answer
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If $\{X_{t}:t\in\mathbb{Z}\}$ is strictly stationary, then $Y_{t}:=g(X_{t},\dots, X_{t+m-1})$ is also strictly stationary.

A stochastic process $\{X_{t}:t\in \mathbb{T}\}$ is strictly stationary if its finite dimensional distribution is stable under any time shift: for any $r\in\mathbb{T}$, any $t_{1},\dots, t_{n}\in\...
JacobsonRadical's user avatar
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Conditional mean of an ARMA($p,q$) process

This question is about a paragraph in The Analysis of Time Series: An Introduction with R (7th Edition) by Chatfield and Xing. I quote Section 12.1, p. 135: In particular, suppose that $X_t$ follows ...
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Stationary distribution for a function of Markov process

Suppose $E$ is a locally compact Polish space, and $(X_t)_{t\ge 0}$ is a Markov process on $E$ with a Feller transition semi-group $P_t:C_b(E)\to C_b(E)$ with a stationary or even ergodic probability ...
JY0's user avatar
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1 answer
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Example of a Markov Chain which is stationary but not reversible [closed]

There are a few questions on this site which ask for an example of a stationary but not a reversible Markov Chain. However most of them go straight to talking about irreducibility. Is there an ...
attack's user avatar
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Stationary Ornstein-Uhlenbeck process - Brownian motion for $t<0$?

I am trying to understand the various definitions of stationarity of the OU process and I can't reconcile a step below. Let me summarize what I know and did. So the OU process is defined by the SDE $$ ...
Andras Vanyolos's user avatar
3 votes
1 answer
65 views

$\beta$-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 ...
Sizhe Ding's user avatar
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1 answer
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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 ...
Viktoria 's user avatar
2 votes
1 answer
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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 ...
René Quijada's user avatar
2 votes
1 answer
30 views

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 ...
amsmath's user avatar
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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, ...
Raidriar's user avatar
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1 answer
206 views

Is an AR(1) process with Bernoulli errors mixing or ergodic?

Before the $\text{AR}(1)$ model, first look at a simpler example $$y_t=\rho^t y_0+\epsilon_t$$ where $0<\rho<1$ and $\epsilon_t\overset{\text{i.i.d.}}{\sim} \text{Bernoulli} \left(\frac{1}{2} \...
Jack's user avatar
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0 votes
1 answer
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Proving $S_m=\sum\limits_{j=1}^m{\theta^j X_{n-j}}$ converges in mean square as $m\to \infty$

Suppose that $\{X_t, t = 0, \pm1,\dots\}$ is is stationary and that $|\theta|<1$. Show that for each fixed $n$, the sequence $S_m=\sum\limits_{j=1}^m{\theta^j X_{n-j}}$ converges in mean square as $...
dienhosp3's user avatar
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1 vote
1 answer
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Doubt related to weakly stationary stochastic processes.

The formal definition I've been given relating to weakly stationary processes follows: Definition. A stochastic process $X = (X_t, t \in T)$ is weakly stationary if $\forall t \in T, E[X_t^2] < \...
xyz's user avatar
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1 vote
1 answer
236 views

Let ${Y_t}$ be a stationary process with mean zero and let $a$ and $b$ be constants. Prove $X_t$ is a stationary

Let ${Y_t}$ be a stationary process with mean zero and let $a$ and $b$ be constants. Show that $X_t = Y_t - Y_{t-1} - Y_{t-12} + Y_{t-13}$ is stationary. I encountered this issue when working on a ...
dienhosp3's user avatar
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W.s.s. Gaussian process in LTI, probability of the output signal

Let {${X(t); t\in ℝ}$} be a wide sense stationary Gaussian process with mean $\mu_X = 1$ and power spectral density $$S_X(f) = \begin{cases} 1, \ \text{if} \ |f| < 5; \\ 0, \ \text{otherwise}. \...
AANICR's user avatar
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1 answer
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linearized operator for ODE system

Consider a system of reaction-diffusion equations where we write as $$ \begin{cases} u_t=\Delta u+f(u,v),\\ v_t=\Delta v+g(u,v) \end{cases} $$ In vector form, we also have $U_t=F(U)$ where $U=\begin{...
79999's user avatar
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0 answers
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Lower bound for probability of birth-death process at 0

Consider a birth-death process with birth transition rate of 1 and death transition rate of $r + \gamma$ at every state $r \in \mathbb{N}$. Can we come up with an lower bound efor the steady-state ...
Alireza Amani's user avatar
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1 answer
46 views

Recovering Entropy Rate Property for Random Walk Across Unweighted Graph

I am self-studying Information Theory and came across this problem concerning the entropy rate of a random walk across this graph. For all logarithms, I am working in base 2. $$\mu=(3/16, 3/16, 3/16, ...
Philo-Sophism's user avatar
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0 answers
21 views

Stationary Distribution Defined Differently In a Textbook (Clarification on Definition of subscripts of $mu$)

for a stationary distribution, I am familiar with the interpretation that the stationary distribution: $\mu$ is the distribution satisfying: $$\mu=\mu*P$$ However, there is a different ...
Philo-Sophism's user avatar
1 vote
0 answers
19 views

Invariant event of stationary processes

Let $(\Omega, \mathcal{F},P)$ be a probability space and let $(X_n)_{n\geq 0}$ be a real-valued stationary process. Let $B\subset \mathcal{R}$, then why is $$ A:=\{X_n \in B,\text{ i.o.}\} $$ ...
Stephen_lamb's user avatar
3 votes
1 answer
58 views

Stationary Distribution of $X_{n+1} \sim \mathcal{U}[0,2X_{n}]$

Let the Markov-process $(X_n)_{n\in\mathbb N}$ be defined by $$ X_{n+1} \sim \mathcal U[0, 2X_{n}], $$ where $\mathcal U$ is the uniform distribution and $X_0$ is positive. I'm interested in finding ...
P.S. Dester's user avatar
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1 vote
0 answers
55 views

Autocorrelation and Power Spectral Density for Wide-Sense Cyclostationary Processes

I am reading in a book called "Understanding Jitter and Phase Noise" and came across the following equation and need a little help to understand his justification for a certain step in the ...
Mohamed Osama's user avatar
1 vote
1 answer
32 views

Minorisation and Coupling in probability theory.

In probability theory, when studying the convergence of a stochastic process to equilibrium minorisation conditions can be exploited (see for instance Assumption 2.1. of https://www.sciencedirect.com/...
Monty's user avatar
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2 votes
1 answer
50 views

Understanding the Independent blocks construction of an ergodic process

I've been wanting to understand the construction of an ergodic process $(Z_m)_{m \in \mathbb N}$ from a non-ergodic process $(X_m)_{m \in \mathbb N}$ for a long time. For this, consider the following ...
Fam's user avatar
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0 votes
1 answer
227 views

AR(2) covariance stationarity

I have equation given like this: $y(t)$ = $\beta_0$ + 0,5($y(t-1)$+$y(t-2)$)+$u(t)$, $u(t)$ ~ iiN(0,sigma^2) In AR(1) we look at the $\beta_1$ before $y(t-1)$ and if its $|\beta_1|<1$ its ...
yoshe's user avatar
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0 votes
0 answers
140 views

Proving autocorrelation is periodic for a wide sense stationary

Let $ \left\{ X\left(t\right)\right\} _{t\in R} $ be a WSS process. There is a number $T\in \mathbb{R} $ such that $R_X(0)=R_X(T) $ where $R_X$ is the autocorrelation function. Prove that $R_X$ is a ...
Danny Blozrov's user avatar
3 votes
0 answers
175 views

Intuitively, why does the spectral gap control the speed of convergence to equilibrium.

Is there an intuitive way to understand the following principle : Given a Markov Process $X_t$ with generator $L$, why does the spectral gap of $L$ control the speed of convergence to equilibrium for $...
Monty's user avatar
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1 vote
0 answers
44 views

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 ...
σκουλήκι's user avatar
2 votes
0 answers
23 views

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 : ...
Sammy Apsel's user avatar
1 vote
0 answers
55 views

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 ...
Milly's user avatar
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0 answers
57 views

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 ...
Anonymous's user avatar
  • 108
2 votes
0 answers
40 views

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_{...
Rüdiger's user avatar

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