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.
Questions tagged [stationary-processes]
447 questions
10
votes
1 answer
Why Brownian motion is a stationary process
I found out that I simply can't rigorously prove that Wiener process is a stationary process, i.e. its finite-dimensional distributions don't change under shift in time. Let $W_t$ be a Wiener process:
$W_0=0$ a.s.
for $0\le t_0\le \ldots \le t_n$…
Mushtandoid
- 551
- 5
- 13
8
votes
1 answer
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
\begin{equation}
~~~~~~~~~~~~ \partial_t \rho =…
8
votes
1 answer
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 distribution in $[-\frac{1}{2},\frac{1}{2}]^d$ and…
Velobos
- 2,190
8
votes
1 answer
Asymptotic joint distribution of sample mean and sample variance
I have a square integrable strictly stationary time series $(r_t)$. Suppose that $(r_t)$ satisfies certain conditions such that
$$\sqrt{T}(\bar{r}_T-\mu_r) \rightsquigarrow N(0,\sigma_1^2)$$
$$\sqrt{T}(s_T^2-\gamma_r(0)) \rightsquigarrow…
Calculon
- 5,843
8
votes
1 answer
Can a reducible Markov chain have an unique stationary distribution?
I know for irreducible and positive recurrent Markov Chain there exists an unique stationary distribution. For Markov Chain with several communication classes (example C1, C2) there exist stationary distributions (linear combination of $ \pi $1 and…
Kathy
- 97
7
votes
1 answer
For finite Markov Chain, time average distribution is always a stationary distribution?
Given some finite state space $\Omega\equiv\{\omega_1,\ldots,\omega_n\}$ be given, and let any Markov chain $\{X_t\}$ with $n\times n$ transition matrix $A$ on $\Omega$ be given. I would like to know if there are standard results out there I can…
fltfan
- 91
- 6
7
votes
1 answer
Please can someone help me to understand stationary distributions of Markov Chains?
I'm currently trying to understand (intuitively) what a stationary distribution of a Markov Chain is? In our lecture notes, we're given the following definition:
This was of little benefit to my understanding, so I've tried searching online for a…
M Smith
- 2,757
6
votes
1 answer
Strictly stationary exponential Ornstein-Uhlenbeck process?
Can one define the initial value of the exponential Ornstein-Uhlenbeck process $r$, defined by
$$r(t) = e^{y(t)}\quad\text{with}\quad dy(t) = k(θ −y(t)) \mathrm dt+\sigma \mathrm dW(t),$$
such that the process is strictly stationary? I would guess…
user43130
- 93
6
votes
1 answer
Show that $X_{t}:=\alpha X_{t-1}+\epsilon_{t}$ is strictly stationary for $|\alpha|<1$ and $\epsilon_{t}$ i.i.d$~\sim N(0,\sigma^{2})$.
The title can be shortened to "prove that $AR(1)$ processes are strictly stationary when $|\alpha|<1$". This has been discussed many times on MSE and Cross Validated, but I found no mathematical proof of why it is strictly stationary.
For…
JacobsonRadical
- 6,128
6
votes
0 answers
The conditions for $X_t=\alpha +\beta X_{t-1} + N_t$ to be wide sense stationary
Assume that a discrete-time process is given as below:
$$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 parameters of the process.
(a) What conditions should be…
Prinko
- 61
6
votes
1 answer
Proving stationarity of AR(1) process
I would like to prove that the AR(1) process: $X_t=\phi X_{t-1}+u_t$, where ${u_t}$ is white noise $(0,\sigma^2)$ and $\vert\phi\vert<1$, is covariance stationary. One requirement is that $\mathbb{E}(X_t)$ is a constant (in this case should be…
SemiMetrics
- 61
6
votes
2 answers
Does the power spectral density vanish when the frequency is zero for a zero-mean process?
A wide-sense stationary random time series $\zeta(t)$ is characterized by its mean value and its autocovariance function, which in the Wiener–Khinchin theorem is equivalent to the Fourier transform of the power spectral density …
P T
- 176
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5
votes
2 answers
Confusion in stationary process and stationary distribution
I have just started studying statistics, and have no background in this field(though I have a descent enough mathematical background).
I was studying about stationary distributions and stationary processes, and the book I am reading says that
A…
user101874
- 53
5
votes
2 answers
Is $\{\sin(\omega n), n \geq 1\}$ a strictly stationary process?
Let $X(t)=\sin(\omega t)$, where $\omega$ is is uniformly distributed R.V. on $[0,2π]$. Let $X_n=X(n)$, is $\{X_n,n \geq 1\}$ a strictly stationary process?
I've calculated that the distribution function of $X_n$…
DANG Fan
- 179
- 1
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5
votes
1 answer
Function of stationary processes
Suppose we have stationary processes $X_1(t), X_2(t),..., X_n(t)$ and let $f_t(X_1(t), X_2(t),..., X_n(t))$ be a continuous function of these stationary processes. Will $f_t(\cdot)$ also be stationary in general?
PS. Not home work, for understanding…
triomphe
- 3,948