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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14 views

Operator theory proof for sub-multiplicativity property of distance to stationarity

For a Markov chain on discrete space $\mathcal{X}$ with transition kernel $P$ and stationary distribution $\pi$, we can define the following distance to stationarity: $$\bar{d}(t) = \frac{1}{2}\max_{(...
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8 views

Lumped Markov chain

Let $X_{n}$ be a Markov chain on a finite set $S$ of states, with transition matrix $P = (p_{s,s'})_{s,s' \in S}$ and initial distribution $p = (p_{s})_{s \in S}$. Let $C = \{ C_0, \ldots, C_r \}$ be ...
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22 views

SDE: conditions for the process to be normal and/or stationary

We have random process $X(t)$ satisfying the following SDE: $dX(t)=A(X(t))dt+B(X(t))dW(t)$, with $W(t)$ - Wiener process. Does somebody know sufficient/necessary conditions on $A$ and $B$, that the ...
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21 views

Let $(X_n)_{n\geq 0}$ be markov$(\pi, P)$ . Show that for any $m\geq 0$ that $Y_n=X_{m+n}$ is markov $(\pi, P)$.

Let $(X_n)_{n\geq 0}$ be markov$(\pi, P)$ and suppose that $\pi$ is an invariant distribution for $(X_n)_{n\geq 0}$. Show that for any $m\geq 0$ the process $(Y_n)_{n\geq 0}$ defined by $Y_n=X_{...
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13 views

transform variance to constant

Suppose I have a sequence of data $\{x_t\}_1^N$ whose variance is a function of time, $\sigma^2(t) = \sigma_0^2 *t$, where $\sigma_0^2$ is a constant. How can I transform the variance of the entire ...
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2answers
52 views

Given a discrete time Markov chain with three states $\{1,2,3\}$ and the transition matrix given: [closed]

Let $$A=(a_{ij})_{3 \times 3}=\begin{pmatrix} 0.5 & 0.5 & 0 \\ 0.5 & 0 & 0.5 \\ 0 & 0.5 & 0.5 \end{pmatrix}_{3 \times 3}$$ where $a_{ij}=Pr\{X_t+1=j | X_t=i\}$ where $X_t$ ...
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7 views

inverse filtering time series

Let $(z_t)_t$ be a weakly stationary process and $A(z) = 1 - \phi ^{-1}z$ where $|\phi| \leq 1$. The inverse filtering theorem states that if the coefficients of the polynomial are summable and if $$ ...
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1answer
28 views

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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22 views

Stationary processes

I'm still learning English so I'm already sorry for what's going to happen. I don't know if I am right about the strictly stationary. Let's suppose a process is strictly stationary: does that mean $...
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37 views

Probability Flow and Markov Chains

There is an exercise that I'm trying to do: $\pi$ is a stationary distribution if and only if $\pi \textbf{1} = \textbf{1} $, and $F(A,A^c)=F(A^c,A)$, where the “probability flow” from a set $...
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25 views

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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17 views

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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5 views

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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16 views

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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1answer
34 views

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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16 views

Autoregressive process with random walk perturbation.

Suppose we have an autoregressive process, $$y_t=\phi y_{t-1} +u_t$$ where $|\phi|<1$. If $u_t$ is an i.i.d random variable this process is stationarity. What if $$u_t=u_{t-1}+g+\epsilon_t$$ where $...
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15 views

gaussian white noise implies gaussian arma process

An ARMA(p,q) process is a (weakly) stationary process $x_t=\sum_{i=1}^p\phi_ix_{t-i}+z_t+\sum_{j=1}^q\theta_j z_{t-j}$ where $z_t$ is white noise. Lets assume that $z_t$ is Gaussian white noise. ...
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1answer
22 views

Show that $𝑋_𝑡=𝑍_1+⋯+𝑍_𝑡$ where $\{𝑍_𝑡\}_{𝑡\geq1}$ is not stationary.

$X_t=𝑍_1+\dots+𝑍_𝑡$ where $\{𝑍_𝑡\}_{𝑡\geq 1}$ is a white noise having the following properties : $𝑍_𝑡 \sim 𝑁\left(0,𝜎^2\right)$ $\forall t\geq 1$ $\gamma_𝑍(h)=0\: ∀ h≠0$ $𝑍_𝑡$ ⫫$𝑍_{𝑡+ℎ}...
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19 views

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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1answer
27 views

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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33 views

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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1answer
42 views

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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1answer
289 views

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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0answers
20 views

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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15 views

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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1answer
62 views

What exactly does it mean to multiply a vector by the transition matrix of a Markov process?

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{...
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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 ...
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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, ...
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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-...
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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 ...
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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 ...
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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 ...
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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 $E(X_t) = \cos(ct)E(Y) = 0$ $E(...
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21 views

Questions about ARIMA modelling

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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83 views

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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0answers
58 views

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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23 views

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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16 views

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.
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128 views

Power Spectral density of a Wide sense stationary Random Process

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 ...
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0answers
24 views

How do they simplify the autocorrelation function?

In my book they state that ...the autocorrelation function can be defined as $$r_X(s,t)=\frac{\text{Cov}[X(s),X(t)]}{\sqrt{\text{var}[X(s)]\text{var}[X(t)]}},\tag 1$$ where $X(t)$ and $...
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17 views

Reversible distribution of a CTMC on a graph, where the edges are cut according to another process.

Hi : I am trying to formulate a theorem i know to be true, but i cant find the proof anywhere, could anyone finish my proof or point me in the direction of the theorem stated properly. THEOREM : ...
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0answers
45 views

Intuitive Explanation to a random variable concept

This is from the Wikipedia page on Stationary Processes: Let Y be any scalar random variable, and define a time-series { Xt }, by Xt = Y for all t. Then { Xt } is a stationary time series, for ...
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110 views

Is an AR(p)-process a martingale?

Is an AR(p)-process a martingale? I think it is not, but I don't know how to explain this. The expected value of the martingale must be zero. In the case of an AR(p)-process it isn't but in the case ...
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8 views

Reversible distribution with cuts on the edges

Say We have a Continuous Time Markov Process $x(t)$ on a irreducible state space (which is countable). with unique reversible distribution $\pi$. I.e $\pi$ solves $$\pi_{i}c_{x}(i,j)=\pi_{j}c_{x}(j,i)...
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34 views

Stationary solution of AR(p) [closed]

If $W_t$ is a white noise then, how can I show that $X_t-\phi_1X_{t-1} - \phi_2X_{t-2} - … - \phi_pX_{t-p} = W_t$ has a stationary solution when $\phi(z) = 1-\phi_1z - \phi_2z^2 - … - \phi_pz^p \neq ...
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1answer
67 views

Biased random walk

I have an undirected graph $G$ and when performing a random walk on the graph $G$, I visit a node $u$ with probability proportional to $d_{u}$ where $d_u$ is degree of node $u$. I know there exists ...
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0answers
21 views

Strict stationarity of a process defined as the product of lags of another process

Here is a problem that just occurred to me and that may be novel or interesting, at least I could find no trace on here. I have a short proof by contradiction in mind but it suffers from a limiting ...
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1answer
66 views

The Covariance of a Homogeneous Irreducible Aperiodic Markov Chain

I am trying to understand this following derivation from a book and need some help. Suppose there is a homogeneous, irreducible, aperiodic Markov chain, whose state at time $t$ is $C_t$, and can ...
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1answer
25 views

Why is $\sum_{i<j}^n \mathbb{P}[X_i > \alpha, X_j > \alpha] \leq n \sum_{j=2}^n \mathbb{P}[X_1 > \alpha, X_j > \alpha]$?

Let $(X_i)$ be a sequence of stationary random variables. Why is the inequality $$ \sum_{i < j}^{n} \mathbb{P}[X_i > \alpha, X_{j} > \alpha] \leq n \sum_{j=2}^{n} \mathbb{P}[X_{1} > \alpha,...