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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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 $0$ ...
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 ...
$F(X_{(n-k_n)})\overset{n\to\infty}{\to} 1$ 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 ...