# 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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### Poisson process ,exponential distribution [on hold]

Suppose in a pharmacy there are two cashiers. The service time of the cashier1 follows a exponential distribution with $\lambda_1$ and the service time of the cashier2 follows a exponential ...
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### Show that a random walk on graph $G$ is an irreducible Markov chain and determine its invariant distribution.

Consider an undirected connected graph $G=(V,E)$ with $|V|<\infty$. Let $(X_n)_{n\geq 0}$ be a random walk on $G$, i.e. it is a markov chain that at each time step moves to a neighbour in the ...
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### Are identically distributed variables sufficient for stationarity? if so of what kind?

If I have variables $Y_t, t = 1, 2, ..$ which are identically distributed, is the time series $(Y_t)$ stationary? If so is it weakly stationary or strictly stationary? Bonus question what are ...
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### How to turn a non-stationary AR process into a stationary process? (ARMA modelling)

In my notes it says for non-stationary processes, forecast differences. E.G. For, y(t)= 1.5y(t-1)-0.5y(t-2)+ε(t) *: which is non-stationary as the coefficients sum to 1 Forecast, Δy(t) = 0.5*Δy(t-...
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### Algorithm for producing a Moving Average (MA(q) as in ARIMA) model.

I have a time series $X_t$ and I want to produce an ARMA forecast (without using any automated packages - the purpose of my project is to understand how those work). So far, I have the AR(p) part ...
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### While proving the Stationairity of a Process, I encountered a problem

Here's the question we got as a handout from the teacher. In the part where we find the expected value of Z(t), I don't understand how we arrived at that the expected value as exp(-2λt). I don't ...
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### Linear combination of terms of stationary process is stationary

I'm just learning about stationary stochastic processes and I'm a little confused about one example. If $X = (X_t)_{t \in I}$ is a stochastic process with values in a Borel space $(E, \mathcal E)$, ...
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### Stationarity vs. Weak Dependence

I know what is stationarity (weak and strict form) and I underestand weak dependence. Can someone give me an example of a process that is stationary and changes through time (not constant), but not ...
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### Determine stationarity of time series containing sin of white noise [closed]

Could someone help me determine the stationarity of the the following time series Y? $Z_t$ represents white noise with variance $\sigma^2$. $Y_t = \sin(Z_t) + Z^2_t - Z_{t-1}$ I have tried ...
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### Example of ANY stochastic process (SDE), with reversible distribution

Can anyone provide an example (as simple as they like) of a process $X_t$ on $\mathbb{R}$ solution to $dX=\sigma (X,t)dt+b(X,t)dW$. Where $W$ is a Brownian Motion, and $\sigma$ and $b$ can be any ...
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### Question for covariance stationary process

Given a random variable Y with characteristic function C(w) = E[exp(iwy)] . Let the random process X(t) be defined as X(t)=cos(wt+y). Show that the process X(t) is covariance stationary if C(1)-C(2)=...
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### What is the difference between a weakly stationary process and strictly stationary process?

In some lecture slides I read that the definition of a weakly stationary process is that The mean value is constant The covariance function is time-invariant The variance is constant and I read ...
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### Determining the parameters of a uniform distribution from its autocorrelation function

I have a noise process v(t) which is wide sense stationary (WSS) and uniformly distributed with autocorrelation $R_{vv}(\tau)=c^2e^{-\beta\mid\tau\mid}$ where c =0.1. How would I find $E[v(t)]$, the ...
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### The property of coercivity in stochastic analysis

Given an SDE $$dX_{t}=b(t,X_{t})dt+\sigma (t,X_{t}) dW_{t}$$ With $W$ the Wiener process. I have seen some results that under some assumptions on the coefficients $b,\sigma$ such as : i) ...
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I know that given a stationary distribution and 2 state transition matrix that $\begin{pmatrix} \Pi _{1} & \Pi _{2} \end{pmatrix}\begin{pmatrix} P_{00} & P_{01}\\ P_{10}& P_{11} \end{... 1answer 44 views ### Ito diffusion: Connection between backward Kolmogorov equation and stationary distribution Suppose we have an Ito diffusion $$dX_t = b(X_t)dt + \sigma(X_t) dB_t, \tag{1}$$ where$dB_t$is Brownian motion. Also assume we know that this diffusion process converges to a stationary ... 0answers 20 views ### Queuing theory for may task in univ A gas station only has one pump for refueling the Pertamax type. The arrival of Pertamax-fueled cars to the gas station follows the process Poisson with an arrival rate of 15 cars / hour. However, ... 0answers 10 views ### Properties of the PDF of a Wide Sense Stationary Process We know that when the process is strict sense stationary, the nth order pdf is time invariant. On the other hand, for Wide-sense stationary processes, we also establish the fact that the the auto-... 0answers 7 views ### nth order strict sense stationarity implications Given positive integers$n, m, n>m$, which sense of stationarity is more restrictive:$n$th order strict sense stationarity or$m$th order strict sense stationarity? Will the more restrictive one ... 0answers 29 views ### Independence of stationary process and it's derivative. Let$X(t)$be a centred stationary gaussian process on the reals, with differentiable sample paths, with covariance function$r(t)$Are$X(0)$and$X'(0)$independent? Why? Are they independent only ... 0answers 30 views ### Multiple Raffle Risk/Reward Optimization Problem This question ponders whether there can be a statistically backed process to determine a distribution of tickets, among a set of raffles, that would probabilistically maximize the net value likely to ... 1answer 78 views ### When is the following process stationary? Let$Y$be a random variable with mean zero and variance$\sigma^2$, and let$c$be a constant. Let $$X_t = Y\cos(ct)$$ When is the process$X_t$stationary? I find$E(X_t) = \cos(ct)E(Y) = 0E(...
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I am estimating this model: But I want to do some analysis of the variables before. In particular, I am interested in fitting some ARIMA models. First, I am doing it for the inflation rate in Mexico. ...
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### Characterization of stationary distribution of diffusion process

Suppose $X(t)$ is a stationary $d$-dimensional Gaussian diffusion process with initial distribution $X(0)\sim\nu$: $$X(t) = X(0) + \int_0^t A X(s) ds + W(t),$$ where $A$ is a strictly negative ...
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### Model fitting and point format

Suppose we fit the model to the n observations (y1, x11, x21), ..., (yn, x1n, x2n): yi = b0 + b1*x1i + b2*x2i + e_i for i = 1, ... n, and where all e_i are iid as a normal random variable with mean ...
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### Relation between spectral measure and sample paths of Ornstein-Uhlenbeck process

The spectral measure of the Ornstein-Uhlenbeck process is absolutely continuous w.r.t the lebesgue measure, and has Cauchy density. Since this density does not have any moments, can we say that the ...
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### How does second order stationarity imply first order stationarity?

If a process is second order stationary ie joint pdf is independent of absolute time, how can it be shown that it is first order stationery as well i.e. first order pdf is independent of time origin.
Let $X(t)$ be a Wide Sense Stationary(WSS) Random Process with the Auto-Correlation function defined as $R_X(\tau) = E[X(t)X^*(t-\tau)]$ . The Power Spectral Density $S(f)$, is defined as the ...