# 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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### Why is the fourier transform of the autocovariance function only integrated from $-\frac{1}{2}$ to $\frac{1}{2}$?

in several time series texts like Shumway, the Fourier transform of the autocovariance function is integrated only over frequencies from $-\frac{1}{2}$ to $\frac{1}{2}$. I do not understand why it ...
1 vote
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### Is the product of independent white noise also white noise?

Assume that the random vector $(u_t, v_t)$ is sampled iid over time and that $E[u_t v_t] = 0$. We also assume that $E[u_t] = E[v_t] = 0$ and that $E[u_t^2] = \sigma^2_u$, $E[v_t^2] = \sigma^2_v$. ...
1 vote
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### Markov chain in a closed interval

Let $I=[0,1]$, $\varepsilon>0$ and $(X_n)$ of i.i.d. random variables (copies of X, with $supp(X)\subseteq (-\varepsilon,\varepsilon)$ and $\mathbb E(X)>0$). Now, for each $\lambda>0$, we ...
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### The frequency function for $Y_t-18=0.4X_t+0.9X_{t-1}+e_t$

I am having trouble finding the frequency function that takes me from $X_t$ to $Y_t$ in the system stated in the title. $X_t$ and $Y_t$ are stationary stochastic processes and $e_t$ is zero mean white ...
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### Distribution and auto-covariance function of a stochastic process

Consider the stochastic process; I see that it is a MA(1) process? From my understanding its covariance function (y_t+h, y_t) is equal to 5/4 when h=0, 1/2 when h=1, and 0 otherwise, is this correct? ...
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1 vote
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### Are discrete-time Markov chains special cases of continuous-time ones?

I am looking for sufficient (and optimally necessary) conditions for a discrete-time Markov chain with uncountable state space to (1) possess a unique stationary distribution with (2) exponential rate ...
1 vote
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### Is stationarity of data necesarry in order to do any statistics?

If we assume we have a stochastic process $X_t$ for which we have, $$\mathbb{E}[X_t] = \mu(t)$$ $$\operatorname{Cov}(X_t,X_s) = \gamma(s,t)$$ where the dependence of the functions on $s,t$ are non-...
1 vote
I consider a one-dimensional SDE $$\mathrm d X_t = F(X_t) \mathrm d t + \sigma(X_t)\mathrm d B_t$$ where $\mathbf P^{X_0} = \mu$ is the starting distribution, $F, \sigma: \mathbf R \to \mathbf R$. ...