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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14 views

Power spectral density of not wide sense stationary

For the random variable $w$ with probabilities $P[w=0]=1/4,\ P[w=1]=3/4$, the random process $X(w,t)$ is given $$X(0,t)=\cos(2\pi t),\quad X(1,t)=\sin(2\pi t).$$ The autocorrelation function of $X(t)$ ...
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34 views

About the ergodicity and stationarity (or lack of) of a particular type of stochastic process

The process in question is a point process. I am mostly interested in the 2-D case but I will accept answers even if it applies to the 1-D case only. The process is defined as follows, on the real ...
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22 views

Fourier-type decomposition for a weakly stationary process

From Wikipedia, there exists a stochastic process $\omega _{\xi }$ with orthogonal increments such that, for all $t$, the weakly stationary process $X_t = \int e^{- 2 \pi i \lambda \cdot t} \, d \...
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12 views

Auto-covariance matrix of $m$-dependent sequences

Suppose that $\{X_t\}_{t\in\mathbb{Z}}$ is a strictly stationary and $m-$dependent (with $m\in\mathbb{N}$) sequence of random variables. Is $\{X_t X_{t+h}\}_{t\in\mathbb{Z}}$ (for fixed $h\in\mathbb{...
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30 views

Auto-Correlation of a discrete Markov chain

goal: I am trying to derive a formula for the autocorrelation of a arbitrary discrete Markov chain, that has reached stationarity. My question is very similar to the ones in question1 and quetsion2. I ...
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18 views

strong stationary time

I would like to prove this two definitions of strong stationary time is equivalents. However, one of the sides has already managed to resolve. I want to prove: $\displaystyle P\{X_k = i |T \le k\} = \...
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37 views

Stationary, Martingale difference sequence, Ergodic process

Does the sequence $\{x _{t}\}_{t=1}^{n}$ satisfy the following conditions if it is generated from a normal distribution with a mean of 0 and a standard deviation of 3? *Stationary? Yes, because the ...
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24 views

strong uniform time

I would like to understand the definition below. A strong uniform time $T$ is a randomized stopping time for $\{X_n\}_{n \ge 0}$, where $X_n$ is a markov chain, such that (i) $\displaystyle P\{X_k = ...
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1answer
24 views

Is the product, of two strictly stationary independent processes, strictly stationary? [closed]

I'm a little unsure if we have $(X_t)$ and $(Y_t)$ are two independent and strictly stationary processes would $(Z_t)$ given by $Z_t = X_tY_t$ also be strictly stationary? I have had no problem ...
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33 views

Why is a one-dimensional brownian motion intrinsicly stationary, but a two-dimensional isn't?

In my script it says, a one-dimensional brownian motion intrinsicly stationary, but a two-dimensional isn't. I can't understand why. Can someone please help me? (Instrinsically stationary means that ...
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1answer
31 views

Why has a weak stationary gaussian process to be strictly stationary?

I'm new to stachastic processes and unsure about stationarity. In my script it says a weak stationary gaussian process is always a strictly stationary process as well. I have an example of a weak ...
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12 views

Survival analysis: Treatment "parameter" for Weibull

I got following question: We let $T^*_1,...,T^*_n$ be independent survival times for n patients and we let $X_i\in\{0,1\}$ indicate if the i-th patient is treated $(X_i=1)$ or not $(X_i=0)$. We are ...
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42 views

Asymptotic expected number of visits to state j from stationary distribution of continuous Markov Chain

My question is as follows: I have a continuous Markov chain with stationary distribution $\pi_j$. I want to calculate the expected number of visits to state $j$ as $t\to\infty$. My intuition suggests ...
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2answers
61 views

To prove that a process is strictly stationary

I have been given $ \{X_n;n>0\}$ as a sequence of i.i.d random variables and we another sequence $ \{Y_n;n>0\}$ defined such that $$Y_n=X_n+aX_{n-1}$$ where a is a real constant I need to show ...
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39 views

Stationary distribution for a countable state space Markov Chain

I am working on a countable state space, irreducible, aperiodic Markov chain with not very easy expresable terms in its stochastic matrix. My goal is to prove that it has got a stationary distribution....
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93 views

Uniqueness of stationary distributions in continuous time

Question Consider a Markov process $\{X(t)\}_{t \geq 0}$ with state space $$S = \{0, 1, 2, 3\}$$ and generator matrix $$Q = \begin{pmatrix} -q_0 & 2 & 0 & 0\\ 2 & -q_1 & 4 & 0\\...
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1answer
112 views

Finding the stationary distribution from the generator matrix

Question Consider a Markov process $\{X(t)\}_{t \geq 0}$ with state space $$S = \{0, 1, 2, 3\}$$ and generator matrix $$Q = \begin{pmatrix} -q_0 & 2 & 0 & 0\\ 2 & -q_1 & 4 & 0\\...
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1answer
40 views

Proving that $\{X(t)=tA\}$ where $A\sim \mathrm{Uniform}(2,9)$ is not strict-sense stationary

Definition: We say that $\{X(t)\}$ is a strict-sense stationary process if the joint distribution of any set of samples does not depend on the placement of the time origin, or in other words: $$ \...
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22 views

Limit theorem for a conditional distribution on stationary mixing fields

For a project, I am looking into central limit theorems for stationary mixing random fields. I found many results for the asymptotic distribution of partial sums, such as that of Bolthausen (1982). ...
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1answer
41 views

Stationary increments definition

Let $X_t$ be a stochastic process on the reals. I was under the impression the definition of stationary increments was that $X_{t+s}-X_{s}$ has the same distribution as $X_{t}-X_{0}$ for all $s,t$. I ...
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28 views

Inconsistent solutions when computing the stationary distribution for small Markov chain

I have a small discrete Markov chain with the following transition matrix: $$ P = \begin{bmatrix} 0 & 0 & 1-r & r \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ r & 1-r &...
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28 views

Numerical Solution of an SDE in the long time limit. ( Reference?) Stability?

Consider an SDE (say scalar valued, with some initial value) \begin{equation*} dX_t=b(X_t)dt+\sigma(X_t)dW_t. \end{equation*} Fix a time step $\Delta t>0$, and consider its Euler-Maruyama ...
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1answer
39 views

Can I make a Poisson process by combining several non-Poisson processes?

I'm interested to know if I can split a Poisson process into non-Poissonian sub-processes or not. Or equivalently, I want to know if an (ensemble) Poisson process can be produced by other-none ...
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0answers
32 views

Continuous time law of large numbers for a stationary process?

Suppose $\{X(t)\}_{t>0}$ is a continuous time weakly stationary stochastic process, and suppose it has $0$ mean. Then $\gamma(t,h):=\text{Cov}[X(t),X(t+h)]=\mathbb{E}[X(t)X(t+h)]$ as a function is ...
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1answer
69 views

Is an $\mathrm{ARMA}(1,1)$ process with an $\mathrm{ARCH}(1)$ innovation strictly stationary?

During some self studies of time series and ARCH processes, I thought of the following example. Given an $\mathrm{ARCH}(1)$ process $Z_t$, \begin{align*} Z_t &= e_t\sqrt{h_t}\\ h_t&=\alpha_0+\...
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34 views

Existence of strictly stationary and ergodic solution

I am looking for a sufficient condition so that the following equation $y_{t}=\alpha_{R_{t}}+\beta_{R_{t}}y_{t-1}+Q_{R_{t}}e_{t}$, where $R_{t}=1$, if $y_{t-1}\leq r_{1}$, $R_{t}=R_{t-1}$, if $r_{1}&...
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1answer
39 views

A statistic to capture the degree of mean reversion

Given a realization of a stochastic process, $x_{t_1}, x_{t_2}, \ldots, x_{t_n}$, is there a simple statistic that captures the degree to which the stochastic process is mean reverting? For example, ...
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40 views

The stability of a gradient flow ( discrete scheme, JKO, proximal point, reference request)

Define a free energy functional on the space of probability densities ( on $\mathbb{R}^d$, denoted $\mathcal{P}(\mathbb{R}^n)$) $$E(\rho):=\int_{\mathbb{R}^d} f(x) \rho(x) dx+\int_{\mathbb{R}^d} \rho(...
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1answer
37 views

For a stationary and ergodic discrete stochastic process $\{X[n]\}_t$, $X[n]$'s are Independent and identically distributed?

I can find some materials that say "if $X[n]$'s are independent and identically distributed (IID), then a random process $\{X[n]\}_n$ is an ergodic process." I think the converse does not ...
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59 views

The Lojasiewicz inequality

I am reading this paper https://projecteuclid.org/journals/differential-and-integral-equations/volume-26/issue-5_2f_6/Convergence-to-equilibrium-for-discretizations-of-gradient-like-flows-on/die/...
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75 views

Langevin equation and convergence to stationary solutions. Free energy. SDE. FPE.

Let $f\geq 0$ be Lipschtiz. The overdamped Langevin equation \begin{equation}\label{eq overdamped Langevin SDE} dX=-\nabla f(X)dt+\sqrt{2} dW_t \end{equation} with Kolmogorov forward equation \...
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33 views

The conditions for $X_t=\alpha +\beta X_{t-1} + N_t$ to be wide sense stationary

Assume that a discrete-time process is given as below: $$X_t=\alpha +\beta X_{t-1} + N_t$$ where $N_t$ is an i.i.d process with mean 0 and variance $\sigma^2$ and $\alpha$ and $\beta$ are the ...
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38 views

Limit of stationary distribution (or invariant measure)

$\{X_{i}(\Delta )\}_{i=1}^{\infty }$ is an ergodic process with state space $% \mathbb{R}$, and $X_{i}(\Delta )$ depends on a parameter $\Delta \in (0,1]$. $X_{i}(\Delta )$ has a unique stationary ...
2
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1answer
38 views

Showing a certain time series is stationary

I have been trying to solve the following problem, but have not been successful yet. I was hoping anyone could nudge me in the correct direction. Let $\{ w_t; t = 0,1,\ldots \}$ be a white noice ...
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13 views

Limiting distribution for a periodic single-server queue

Consider a $M/M/1$ queue with a constant arrival rate $\lambda$ and service rate $\mu$ with $\lambda < \mu$. We know that in this case the limiting distribution exists and it is a geometric ...
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11 views

Mathematical & Real-life explanation of a regression operator

Good day! I'm reading about autoregressive models and would appreciate some help in understanding/interpreting the maths from the Explanation of autoregressive model with respect to linear filter (...
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12 views

Optimal sample length - optimal experiment planning

I have a measurement signal with a length of, say, 1 million samples. On the signal, I use digital signal processing techniques to obtain information that interests me. I noticed that I don't have to ...
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30 views

Does this process for generating an integer sequence have a formal name?

Suppose I have $n \in \mathbb{N}$ bins, each having unit volume. Let $a_1,a_2,\ldots,a_{n-1} \in \mathbb{R},\;a_k \geq 1\;\forall\;k$. An integer sequence is generated as follows: A man shows up with ...
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81 views

Convergence in probability of conditional second cross-moment for bivariate stationary process

Let $(X_t,Y_t)_{t\in\mathbb N}$ be a bivariate real stationary process, and let $\mathcal F_t:=\sigma(Y_s :s\leq t)$ be the filtration generated by $Y_t$. Assuming the following convergence result $$\...
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17 views

Markov chain discrete waiting

Let a finite Markov chain with discrete time have a stationary distribution $\mu$ and all $\mu_i$ are positive. If initial state is matched with $\mu$, how to prove that average time to reach some ...
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1answer
52 views

Stationary Distribution of CTMC given a rate matrix Q

For c, I understand that from state 1, the chain will either stay at 1 or go to state 2, since the chain is reducible. I am wondering if the stationary distribution of $\pi_2$ is $1/2$ or if the $\...
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17 views

Does a non-stationary source has an entropy rate $H_X$?

Let $ \{X_k\}_{k\in \mathbb{N}_+} $ be a source and we assume that $H(X_k )<∞$ for all $k$. Then we define the entropy rate of an information source, which gives the average entropy per letter of ...
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24 views

Finding the correlation function of the output process at zero and checking if it is WSS

Assume two systems for which the following differential equations hold between their input and output signals. $$a \dfrac{dv(t)}{dt}+b v(t)=x(t)$$ $$\dfrac{dy(t)}{dt}=v(t)u(t)$$ Also, assume that the ...
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2answers
56 views

Invariant measures that aren't reversible (or vice versa)?

It seems like every discussion of reversible Markov chains assumes that the measure is invariant. A reversed chain has the transition probabilities $p'$ satisfying $$ p'_{xy} = \frac{\pi(y)}{\pi(x)} ...
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1answer
52 views

no stationary distribution with none-vanishing limit of transition probability

We know that irreducible Markov chains can be separated into the two cases: (1) All limits of transition probabilities vanish: $lim_{n\rightarrow\infty} p_{ij}^{(n)} = 0$ for all i,j in state space S. ...
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1answer
26 views

Binomial Processes to find probability

Consider a series of independent coin shots with the possibility of $p$ coming head in any shot. Let $Y_n$ be the numbers of heads in first $n$ consecutive shots of the coin. In this case find the ...
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20 views

Stationarity of a process

I have a process: $$\epsilon_t=\nu_t\sqrt{\alpha_0+\alpha_1\epsilon^2_{t-1}}$$ where $\nu_t$ is a white noise with variance equal to $1$. My textbook says that the conditional variance of $\epsilon_t$ ...
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9 views

stationary vector for an unbreakable markov chain with period 3

I need to find an unbreakable markov chain with period 3 on all the natural numbers such that it's stationary vector $\Pi =(\pi_0,\pi_1,\ldots)$ follows: $\pi_1 = \pi_2 = 1/3$ my attempt was that $0$ ...
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1answer
146 views

Convergence of Stationary random variables

We have a stationary sequence of random variables $X_{j}:j\geq 0$ and let $D$ be a Borel subset of $\mathbb{R}^{d}$. For each n, let $Y_{n}$ be the number of indices $i \in \{0,1, \ldots, n-d\}$ such ...
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45 views

$F(X_{(n-k_n)})\overset{n\to\infty}{\to} 1$ for time series?

Let us have a stationary time series $X=(X_t, t\in\mathbb{Z})$ following e.g. AR(p) model (i.e. there are $a_1, \dots, a_p$ such that $X_t=a_1X_{t-1}+\dots + a_pX_{t-p} + N_t$ where $N_t$ are some iid ...

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