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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12 views

Stationary Distribution of a Stochastic Processes

A Markov Chain with states 0,1,... has transition probabilities $$p_{jk}=e^{-a} \sum_{r=0}^k \left( \begin{matrix} j \\ r \end{matrix} \right) p^r (1-p)^{j-r} a^{k-r} / (k-r)!$$ Show that the limiting ...
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27 views

What is $var[X(0)+X(0.5)]$, where $X(t)$ is a stationary process? [closed]

So I have that $\{X(t);t ∈ R\}$, with mean $\mu_X = 0.5$ is stationary Gaussian stochastic process and its autocorrelation function is $R_X(\tau)=0.5e^{-\pi\tau^2/4} $ $\forall \tau \in R. \quad$ What ...
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22 views

Variance of AR(2) stationary process

Given $AR(2)$ stationary process $$ y_t = 2 + 0.6y_{t-1} - 0.08y_{t-2} +u_t$$ where $u_t$ white noise from $N(0,4)$ Find $Var(y_t)$ My problem: When I take the variances of left and right I have a ...
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1answer
24 views

p-value for test regarding sample belonging to gaussian process

Suppose we know that $X$ is a stationary mean zero gaussian process with known parameters. Suppose an experiment provides me with a collection of samples $(t_i, x_i)$ for $i = 1, 2, . . . , N$. How ...
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23 views

Coefficients of stationary process

Problem Given an equation $x_t -2.5x_{t-1} + x_{t-2} = u_t$ where $u_t$ is white noise. Does this equation have a solution of form $ x_t = \dots + \alpha_{-1}y_{t+1} + \alpha_0 u_t + \alpha_{1}y_{t-1} ...
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1answer
26 views

Variance of ARMA process?

Problem Given $ARMA(1,1)$ stationary process $$x_t = 0.7 x_{t-1} + u_t + 0.2 u_{t-1} $$ where $u_t$ is white noise, with standard deviation $\sigma(u_t) = 4$ Note, stationarity of $x_t$ implies that $$...
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18 views

Why isn't a time independent autocorrelation function sufficient for Wiener–Khinchin theorem?

The Wiener–Khinchin theorem says that for wide-sense stationary processes. If the Fourier transform of the autocorrelation function exists, then it equals the power spectral density. Wide-sense ...
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28 views

Conditions under which we can characterize the stationary distribution of a Markov chain with continuous states

Suppose I have a Markov chain in a continuous space. Are there conditions under which I can solve for the stationary distribution easily (e.g. in closed form)? For example, suppose my space is the ...
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56 views

Is a part of stationary process is stationary process?

Problem Assume the we are given a stationary process $x_t$. That is $$E(X) = const$$ $$ Cov(x_t, x_{t+k}) = f(k)$$ It is known that $E(x_t) = 4$, $Var(x_t) = 16$ and $Cov(x_t, x_{t-1}) = 4 $. We go ...
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42 views

Square of stationary variable

I'm given a stationary process $x_t$: $$\Bbb E(x_t) = \mu $$ $$Cov(x_t,x_{t+k}) = \xi_k \text{ some function of } k \text{ and not of } t$$ I need to find $ \Bbb E(x_t^2)$ using $\mu$ and $\xi_k$.
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10 views

Covariance of sum random process

Sum random process is defined as: $$ X[n] = \sum_{i=0}^n U[i] \text{ for } n \ge 0 $$ $$ E[U[i]]=0 \text{, } \text{var}(U[i]) = \sigma^2_U \text{ for } i \ge 0 \text{ and } U[i] \text{ are IID} $$ ...
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40 views

How to prove that $\sum\pi_i = \sum\frac{1}{E_iT_i} = 1$ in an irreducible Markov chain with stationary distribution $\pi$?

In Durrett's book Chapter 5 Theorem 4.6, It said that if p is irreducible and has stationary distribution, then $\pi_i=\frac{1}{E_iT_i}$. Where $T_i$ is the first time the markov process returns back ...
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127 views

Question on the proof of Subadditive Ergodic Theorem in Durrett's textbook.

This came up in Durrett's proof of the subadditive ergodic theorem. Let $X_{m,n}$ satisfy the assumptions of subadditive ergodic theorem, which reads $X_{0,m}+X_{m,n} \geq X_{0,n}$ $(X_{nk, (n+1)k})...
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9 views

Autocovariance function of stationary process $Y_t=X_t-X_{t-1}$

Let the $\lbrace X_t\rbrace$ be a stationary process with autocovariancy function $\gamma_X(k)$. Let $Y_t=X_t-X_{t-1}$. I need to find an autocovariance of $\lbrace Y_k\rbrace$ Using $\gamma_X (k)=\...
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16 views

The integral of the log spectral density for a infinite MA process.

Let us consider a MA($\infty$) process: $y_t= \sum_{j=0}^{\infty} b_j \varepsilon_{t-j} $ where $ \sum_{j=0}^{\infty} b_j^2 < \infty \>\>\>\>\>\> $ (1) Given $f(\lambda)$ as ...
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9 views

Invert Fractional Differencing - Fractional Integrating?

I'm using fractional differencing to make a signal stationary and do regression. The output of the regression model now follows the fractional differenced input I gave it. However, I want to invert ...
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41 views

What does it mean for two equations to be 'invariant'?

I have watched a few videos yet I am still having a bit of trouble. I have included the two equations here: It says that the two equations are invariant and must be anchored. I am totally lost on ...
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45 views

Discrete correlation function (sample cross-covariance)

If I start with a continuous correlation function given by \begin{equation} C_{AA}(\tau) =\frac{1}{T} \int_{0}^T d\bar{t} A(\bar{t})A(\bar{t}+\tau) \end{equation} with $\tau < T$. How can I prove ...
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10 views

is it stationary?

$U_t$ is a sequence of i.i.d random variables uniformly distributed in $[0,1]$, let $Z$ be $N(0,1)$ indipendent of $U_t$. Define $Y_t=Z+U_t$, then is $Y_t$ stationary? If $\overline {Y_n}= \frac{1}{n}\...
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161 views

How to show ergodicity on this probability measure.

I am looking at a way of describing an infinite checkerboard where in each tile a random constant matrix of size $d \times d$ is given. Step 1 : introduction Let $z$ a random vector with uniform ...
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17 views

Stationary in stochastic process

Show that a random process which is stationary to order $n$ is also stationary to all orders lower than $n$. I searched about order in stationary but I really didn't understand anything, can anybody ...
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6 views

Detecting when an increasing mean becomes higher than a stationary mean with concentration inequalities

Given T time steps $t=1,..,T$. Suppose at each step $t$ I draw two samples from two different distributions. The first distribution has fixed mean, while the other has mean increasing with $t$, and ...
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37 views

Expected Value of AR(1) Process

The first order autoregressive process is defined in my textbook to be $ε_t=\phi a_{t-1}+a_t$. Assuming the model is stationary I want to find the expected value of $ε_t$. If I understand correctly, ...
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5 views

Is converting stationary regression output back to non-stationary format valid?

Say I want to do some regression. The thing I am interested in modelling is non-stationary. To have a well behaved noise term I differentiate my dependent and independent variables to something ...
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28 views

$Y_{t}$ is centered and stationary process. Spectral density of Y is $f(\lambda)=\sin^{2}(\lambda)*I_{[-1;1]}$

Find spectral density of $X_{t}$ and $\text{Cov}(X_{1}, X_{4})$ if $Y_{t} = \int_{t-1}^{t+1}X_{s}ds$. I think that it is necessary to prove that $X_{t}$ is also stationary. But even if I could do that,...
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13 views

Local stationarity in layman's terms.

I am struggeling with the notion of local stationarity. See for instance section 5 of this paper for an introduction of the notion: https://halshs.archives-ouvertes.fr/halshs-00187875/document. My ...
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85 views

Showing a Poisson process is stationary

Let $\{N_t: t\ge 0\}$ be a Poisson Process with rate $\lambda$ and $T_0$ is an independent r.v. where $$ \Bbb{P}(T_0=1)=\mathbb{P}(T_0=-1)=\frac12$$ If we then let $T_t = T_0(-1)^{N_t},$ how can we ...
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29 views

Time series. Showing that a process is stationary.

So the problem, that I have is the following: Let $\{Y_t\}$ be a stationary process with $\mu=0$ and $a,b$ constant. Define and new stochastic process $X_t = (a + bt)s_t + Y_t$, where $s_t$ is the ...
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34 views

Convergence of AR(1) process

Let $\{X_n : n \ge 0 \}$ be a sequence of i.i.d N(0,1) random variables are define $Y_0 = X_0$ and $Y_{n+1} = \phi Y_n + X_{n+1}$, $|\phi| < 1$. a) Show that $Y_n$ is a Gaussian RV and that $Y_n \...
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14 views

Using Stationary Distribution in the Policy Gradient Theorem Proof

As per Sutton et al. 1998, consider a Markov decision process (MDP), where the state, action and reward at each time $t \in \{0, 1, 2, \dots\}$ are denoted by $s_t \in \mathcal{S}, a_t \in \mathcal{A}$...
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16 views

Does first order stationary random process means that all variables in the collection have the same distribution?

Looking at the defintion of N-th order stationarity, The only way I see a random process to be 1st order stationary is when all its random variables (any time) have same distribution. Am I correct?
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29 views

What are the requirements for $\lim_{h\to 0^+}\int_0^{t}f(X_{s+h})dX_{s+h}=\int_0^{t}f(X_{s})dX_s\tag{1}$ to hold?

Let $X(t)$ be a continuous stochastic process that has increments of fixed length that are strongly stationary, that is $X(t+s)-X(t)$ are identically distributed for any fixed $s>0$. Then I choose $...
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30 views

DTMC with a stationary distribution is positive recurrent

I have an irreducable discrete time Markov chain (DTMC) $(X_n)_{n\geq0}$ with finite state space $\mathcal{X}$. The DTMC has a stationary distribution $\pi$, such that $\pi = \pi P$, where $P$ is the ...
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44 views

Does conditional stationarity of $X_t$ imply that $\lim_{j\to \infty} \mathbb{E}_t[X_{t+j}]$ exists?

Let $X_t$ be univariate stochastic process. Assume $X_t$ is conditionally stationary that is for any $n\in \mathbb{N}$ and time indices $t_1,\ldots, t_{n+1}$, $t_{i}<t_{i+1}$ and shift $\tau$ $$F(...
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1answer
41 views

minimising the square of the expected error in Gaussian time series model

I'm trying to derive the result that $\mathbb{E}(X_{n+h} - m(X_n))^2 = \sigma^2(1-\rho(h)^2)$ where $m(X_n) = \mu + \rho(h)(X_n - \mu)$ but i can't get the same result. I will post my derivation below....
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48 views

Difference between stationary and homogeneous point process

I do not understand the difference between a stationary and a homogeneous point process. The definitions I found are as follows: A process is stationary if the entire configuration of the process is ...
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20 views

Convergence of empirical measure together with an ergodic process

Let $\xi=( \xi_i)_{i\in\mathbb N}$ be a discrete time $\mathbb R$-valued stochastic process. Assume that $\xi$ is stationary and ergodic. Let also $\mu$ be a probability measure on $\mathbb R$ and $(...
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1answer
38 views

If I split a stationary ARMA process into two parts, are they also stationary?

Considering an Auto-Regressive Moving Average (ARMA) model, \begin{equation*} y_k = \phi_0 + \sum_{j=1}^{p} \phi_j y_{k-j} + \sum_{l=1}^{q} \theta_l \varepsilon_{k-l}+ \varepsilon_k, \qquad \text{for}...
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26 views

Maximum Entropy Random Walk on Graph Obtained with Multi-Valued Exponentiation

Let $$ n^s = e^{(\log n + 2 \pi i k)(x+iy)}$$ now consider $$ e^{x\log n + 2 \pi i k_1 + iy \log n - 2 \pi k_2}; k_1,k_2 \in \mathbb{N}$$ Apply this to a relevant sum, zeta for simplicity, and look at ...
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105 views

How to infer the “innate” average speed of a frog?

Let's model the motion of a leaping frog as a stochastic process in time. The only thing we know about this process is that it depends on an hidden parameter $V_\infty$: we want to estimate $V_\infty$,...
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19 views

Invariant distributions of transition matrices

Please note down all invariant distributions for each of the following transition matrices. $\begin{pmatrix} 0&\frac{1}{2}&\frac{1}{2}\\ \frac{1}{3}&\frac{1}{3}&\frac{1}{3}\\ \frac{1}{...
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25 views

Conditions for wide sense stationary processes

I was going through my lectures and read that there are two conditions that need to be fulfilled in order to prove that a random process is stationary. Here a screenshot below: I am wondering. Is it ...
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1answer
32 views

Equality in Conditional Distribution for a Stationary Process

Suppose I have a stationary process $X_t$, defined on a probability space $(\Omega,\mathcal{F},\mathbb{P})$. Then we know that for a finite index set $t_1,t_2,...,t_n$ and a shifted index set $t_1+s,...
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14 views

How to show that an ARMA process with non-Gaussian noise is stationary?

It seems that ARMA process is mostly presented with the assumption that the noise is Gaussian (or at least has finite variance) and then the stationary is presented as a condition on the roots of the ...
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5 views

Stationary increment datasets

I have a stochastic counting process that has stationary increments and am struggling to find applicable datasets that I could use for modeling. (The process is not Poisson.) I know that I could ...
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24 views

General Questions about Wide-Sense-Stationary processes.

I learned that if a random process is WSS, its mean should be constant and the correlation only depends on time difference. Also, I learned that white noise is definitely WSS. I tried to simulate ...
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1answer
73 views

Show ergodicity of $2x\operatorname{mod}1$

Let $(E,\mathcal E,\mu)$ denote the Lebesgue measure space on $[0,1)$, $$\tau(x):=2x-\lfloor 2x\rfloor\;\;\;\text{for }x\in E,$$ $$Y_0:=\lfloor 2x\rfloor\;\;\;\text{for }x\in E$$ and $$Y_n:=Y_0\circ\...
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1answer
42 views

Show that this process is identically Bernoulli distributed, indepednent and stationary

Let $$\operatorname{frac}(x):=x-\lfloor x\rfloor\;\;\;\text{for }x\ge0$$ and $$\theta(x):=\operatorname{frac}(2x)\;\;\;\text{for }x\in[0,1)$$ denote the Bernoulli shift. Now define $$X(x):=\lfloor 2x\...
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79 views

Probability of failure at year i vs Probability of failure up to year i

Assume you have a system with unknown resistance $(R)$ at the design phase, therefore that could be modelled as a random variable, but time-invariant (i.e. the value of the random variable generated ...
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15 views

Variance change in a moving average of infinite order?

Does the variance or auto covariance change in the moving average of infinite order, of a weakly second order stationary time series process. Because if I find the variance of the time series ...

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