# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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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 $$Y_n:=Y_0\circ\... 1answer 42 views ### Show that this process is identically Bernoulli distributed, indepednent and stationary Let$$\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\...
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 ...