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.
416
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Integral of a process is not a wide-sense stationary process
Consider $(X_t, t \geq 0)$ which is $L^2$ continuous and stationary in the wide sense process. Suppose it has derivative in $L^2$ and $EX_t \neq 0$, then for no random variable $\xi$ the process $(\xi ...
- 443
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1
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15
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Recovering Entropy Rate Property for Random Walk Across Unweighted Graph
I am self-studying Information Theory and came across this problem concerning the entropy rate of a random walk across this graph. For all logarithms, I am working in base 2.
$$\mu=(3/16, 3/16, 3/16, ...
0
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0
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18
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Stationary Distribution Defined Differently In a Textbook (Clarification on Definition of subscripts of $mu$)
for a stationary distribution, I am familiar with the interpretation that the stationary distribution: $\mu$ is the distribution satisfying:
$$\mu=\mu*P$$
However, there is a different ...
1
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0
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16
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Invariant event of stationary processes
Let $(\Omega, \mathcal{F},P)$ be a probability space and let $(X_n)_{n\geq 0}$ be a real-valued stationary process. Let $B\subset \mathcal{R}$, then why is
$$
A:=\{X_n \in B,\text{ i.o.}\}
$$
...
- 399
3
votes
1
answer
50
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Stationary Distribution of $X_{n+1} \sim \mathcal{U}[0,2X_{n}]$
Let the Markov-process $(X_n)_{n\in\mathbb N}$ be defined by
$$ X_{n+1} \sim \mathcal U[0, 2X_{n}], $$
where $\mathcal U$ is the uniform distribution and $X_0$ is positive. I'm interested in finding ...
- 881
1
vote
0
answers
33
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Autocorrelation and Power Spectral Density for Wide-Sense Cyclostationary Processes
I am reading in a book called "Understanding Jitter and Phase Noise" and came across the following equation and need a little help to understand his justification for a certain step in the ...
0
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0
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10
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Is a Strict-Sense Stationary (SSS) stochastic process passing through a Linear and Time-Invariant system (LTI) stays SSS?
We know that Wide-Sense Stationary (WSS) process passing through a stable LTI outputs a WSS process.
The stability condition is that the sum of coefficients of the LTI doesn't diverge, and that the ...
1
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0
answers
19
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Minorisation and Coupling in probability theory.
In probability theory, when studying the convergence of a stochastic process to equilibrium minorisation conditions can be exploited (see for instance Assumption 2.1. of https://www.sciencedirect.com/...
- 2,100
2
votes
1
answer
46
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Understanding the Independent blocks construction of an ergodic process
I've been wanting to understand the construction of an ergodic process $(Z_m)_{m \in \mathbb N}$ from a non-ergodic process $(X_m)_{m \in \mathbb N}$ for a long time. For this, consider the following ...
- 825
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1
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45
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AR(2) covariance stationarity
I have equation given like this: $y(t)$ = $\beta_0$ + 0,5($y(t-1)$+$y(t-2)$)+$u(t)$, $u(t)$ ~ iiN(0,sigma^2)
In AR(1) we look at the $\beta_1$ before $y(t-1)$ and if its $|\beta_1|<1$ its ...
- 1
0
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1
answer
56
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Proving autocorrelation is periodic for a wide sense stationary
Let $ \left\{ X\left(t\right)\right\} _{t\in R} $ be a WSS process.
There is a number $T\in \mathbb{R} $ such that $R_X(0)=R_X(T) $ where $R_X$ is the autocorrelation function.
Prove that $R_X$ is a ...
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52
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Intuitively, why does the spectral gap control the speed of convergence to equilibrium.
Is there an intuitive way to understand the following principle :
Given a Markov Process $X_t$ with generator $L$, why does the spectral gap of $L$ control the speed of convergence to equilibrium for $...
- 2,100
2
votes
0
answers
38
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A constant matrix multiplying the Lagevin equation (SDE).
Under some conditions on a probability density $\pi$ on $\mathbb{R}^n$, the overdamped Langevin equation
$$dX(t)= \nabla \log \pi(X(t)) dt +\sqrt{2}dW_t $$
has stationary distribution $\pi$, $W_t$ is $...
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1
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0
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25
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Difference between time reversible and stationary distribution. "watching a movie backwards"
A time homogeneous Markov process on Ω
with semi-group $P_t$
is said to stationary w.r.t a distribution $π$
if
$$∫_{Ω}P_tf(x)dπ(x)=∫_{Ω}f(x)dπ(x), \text{for $f$ bounded measurable}.$$
and reversible ...
- 21
0
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0
answers
15
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Strict stationarity and representation via a measure preserving map
I am reading Chapter 7 of the book
Zhikov, V. V.; Kozlov, S. M.; Olejnik, O. A., Homogenization of differential operators and integral functionals. Transl. from the Russian by G. A. Yosifian, Berlin: ...
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2
votes
0
answers
15
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Expectation of product of 3 samples of WSS Gaussian process
Suppose $X_n$ is a WSS Gaussian stochastic process with(every subsample of the process is jointly gaussian):
$E[X_n]=\mu$ for all n
$R_{xx}[l] $ is the autocorrelation function
I wish to calculate : ...
- 69
0
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0
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22
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Show time series is stationary
I have the following time series $$y_t = (-1)^tx_t$$
whereby $x_t$ is an AR model, $$x_t=\frac{1}{2}x_{t-1}-\frac{1}{3}x_{t-2}+\epsilon_t$$
with $\epsilon_t \sim \text{W(0, 1)}$ , show that $y_t$ is ...
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1
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0
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21
views
Can the Wiener-Khinchin theorem be correctly applied to a periodic sound signal (such as a sine wave)?
The theorem speaks about a wide-sense stationary random process. Is, for example, a sine wave with a period 1/400 s considered a WSS (or, in general, a periodic sound signal with multiple frequency ...
- 109
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0
answers
49
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Power Spectrum Definitions
Consider a stationary random process $x(n)$ with $n = 0, \pm 1, \pm 2,..$
The Autocorrelation function is defined as:
$$R_{xx}(m) := \mathbb{E}[x^*(n)x(n+m)]$$
where the * denotes conjugation.
I want ...
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2
votes
0
answers
23
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Weak convergence of $\sum_{k=0}^n \lambda^k X_{n-k}$ if $X$ is a stationary process
Suppose that $(X_k)_{k \in \mathbb{N}}$ is a stationary Markov chain with state space $\mathbb{R}$ and $0<\lambda<1$. Can we say something about the weak convergence of the sequence $Y_n = \sum_{...
- 21
2
votes
0
answers
35
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Show that a certain stochastic process is not linear
Let $Y=(Y_t)_{t \in \mathbb Z}$ be a stochastic process defined on $(\Omega, \mathcal{F}, P)$. We say that $Y$ is a linear process if:
\begin{equation}\label{I}\tag{I}
Y_t = \sum_{j=0}^\infty \psi_j \...
- 825
2
votes
0
answers
57
views
Ergodic Theorem for Markov Chains and convergence to equilibrium
in the lecture we formulated the ergodic theorem for an irreducible and positive recurrent transition matrix P with stationary distribution $\pi$, stating that for bounded functions $f$, it holds: $$\...
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1
vote
1
answer
22
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Derivation of Autocovariance Function of First-Order Autoregressive Process
In my textbook, the autocovariance of the AR(1) model is derived as such:
$$Y_t=\phi Y_{t-1}+e_t$$
After multiplying both sides by $Y_{t-k}(k=1,2,...)$ and take expected values, you get:
$$E(Y_{t-k}...
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1
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0
answers
46
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Are successive recurrence times ergodic for an ergodic process?
In paper "Successive Recurrence Times in a Stationary Process" by Shu-Teh Chen Moy
(The Annals of Mathematical Statistics, Vol. 30, No. 4 (Dec., 1959), pp. 1254-1257),
it was shown that for ...
1
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0
answers
39
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Spectral representation of a white stationary process
I am trying to better understand the spectral representation of stochastic processes. From the book "Spectral Analysis for physical applications" by Walden and Persival:
The spectral ...
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1
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0
answers
28
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Brockwell & Davis, Time Series Theory and Methods, problem 4.4
If $\{X_t\}$ is the process defined by $$X_t=\sum_{j=1}^n A(\lambda_j)e^{it\lambda_j}$$ in which $-\pi<\lambda_1<\lambda_2<...<\lambda_n=\pi$, and $A(\lambda_1), ..., A(\lambda_n)$ are ...
- 41
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1
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49
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Stationary distribution for random walk on $\{1,2,...,N\}$
This is from Probability: An Introduction, by Grimmett and Welsh.
A random walk moves on the finite set $\{0, 1, 2, . . . , N\}$. When in the interior of the
interval, it moves one step rightwards ...
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0
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13
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How to compute the sample autocovariance with multiple i.i.d. samples.
This question is copied of https://stats.stackexchange.com/q/451404.
It has not been answered yet, however, I put the question here.
Consider two (discrete) samples of any stationary stochastic ...
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2
votes
1
answer
50
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Distribution of bivariate vectors for strictly stationary processes
Consider a strictly stationary process $X_t$, $t\in\mathbb{Z}_{\geq 1}$. Could you help me to disprove the following statement:
"For $t, s > 0$, the bivariate vectors $(X_s, X_t)$ and $(X_t, ...
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2
votes
0
answers
136
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Proof of the stationarity condition of ARMA model
In the book Introduction to Time Series and Forecasting by Peter J. Brockwell and Richard A. Davis, at page 75, there is the Existence and Uniqueness of stationary solution of an ARMA process:
...
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0
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0
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46
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joint distribution of strictly stationary process
Let $\{X_t; t \in \mathbb{Z}\}$ be a strictly stationary process, i.e. the joint distribution function on $\mathbb{R}^n$$F_{t_1,\dots,t_n}(x_1,\dots,x_n) = P\{X_{t_1}\leq x_1,\dots,X_{t_n}\leq x_n\}$ ...
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0
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0
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35
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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 ...
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0
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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
1
answer
93
views
Product of two stationary random processes
Let $X(t)$ and $Y(t)$ be two stationary random processes and $$Z(t)=X(t)Y(t)$$By stationary I mean stationary in the strict sense which is $$F_{X(t_1),...,X(t_k)}(x_1,...,x_k)=F_{X(t_1+h),...,X(t_k+h)}...
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0
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How to determine whether a reduced-form VAR is covariance-stationary or not?
I know how to determine whether a vector is covariance-stationary or not but I do not know how to determine whether a reduced-form VAR is stationary or not. For example, the expression is shown below,
...
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0
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25
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Random Walk Visits a Stationary Set
Let $(X_t(v)) \in \{0,1\}^{\mathbb Z^d}$ be stationary random variables. Suppose that the expected amount of time each site $v \in \mathbb Z^d$ is in state $1$ is infinite:
$$\int_0^\infty X_t(v)\: dt ...
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0
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0
answers
50
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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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0
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11
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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 ...
- 183
0
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0
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17
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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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0
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53
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Is the product of weakly stationary processes also weakly stationary?
Suppose $\mathbf{X}_t\in \mathbb{R}^p$ and $\mathbf{Y}_t\in \mathbb{R}^d$ are two weakly stationary processes. Define $\mathbf{Z}_t = \mathbf{X}_t \mathbf{Y}_t^T.$ Speciffically, if $\mathbf{W}_t = \...
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1
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1
answer
47
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Trend Estimation by Differencing
Consider a linear trend model, i.e.
$$X_t = \mu + \beta\cdot t + N_t$$
We then have
$$(\nabla X)_t = (1 - B)X_t = \beta + (1 - B)N_t$$
where $\nabla$ is the lag-1 difference operator and $B$ is the ...
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0
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0
answers
13
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Given discrete-random process $X\left[n\right]=s^n,\!$ with $s\sim\mathtt{Unif}\left[0,1\right]\!,$ seek trend function of ${\it X}\left[n,\xi\right]$
Problem. Given a discrete-random process ${\it X}\left [ n \right ]= s^{n},\!$ with $s\sim\mathtt{Uniform}\left [ 0, 1 \right ]\!,$ seek the trend function of ${\it X}\left [ n \right ]\!$ or ${\it X}\...
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vote
1
answer
26
views
Given $\Lambda\sim\mathtt{Pois}\left(\lambda\right)$ and a stochastic process $X\left(t\right)=\Lambda\cos 2\pi t.$ Seek trend and covariance function
Problem. Given $\Lambda\sim\mathtt{Pois}\left ( \lambda \right )$ and the stochastic process ${\it X}\left ( t \right )= \Lambda\cos 2\pi t.$ Seek the trend function and the covariance function of ${\...
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1
vote
0
answers
56
views
Is this a correct definition of stationary process?
Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space, let $(E,\mathcal E$) a measurable space. Let $(X_n)_n$ be a succession of random variable from $\Omega$ to $E$. We consider the whole $(...
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2
votes
0
answers
37
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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 ...
- 49
1
vote
1
answer
40
views
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-...
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1
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1
answer
69
views
When do invariant solutions of an SDE exist?
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, $F, \sigma: \mathbf R \to \mathbf R$. ...
- 3,426
0
votes
1
answer
65
views
Determine order of an ARIMA process
My problem: ${X_t}$ is a stationary process where ${X_t={\phi}X_{t-1}+Z_{t}+Z_{t-2} }$ with $Z_{t}$ being the error term aka white noise(0,$\sigma^2$). We are given the process ${Y_t=Y_{t-1}+X_{t}-{\...
- 1
3
votes
1
answer
84
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Why is the Spectral Density of a stationary sequence real?
I am trying to understand (and prove) why the spectral density $\Phi_s \in L^1([- \pi, \pi])$ of a stationary sequence $s = \{s_n\}_{n \in \mathbb{Z}}$ is real.
I wanted to argue via the auto-...
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0
answers
20
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Use impulsive response $h(t)$ to calculate $R_{XY}(\tau)$,$R_{YY}(\tau).$
The process $X(t)$ is wide sense stationary $(WSS)$ with $R_XX(τ) = 3δτ$. It is applied to a linear, time invariant $(LTI)$ system with the following input output relationship:
$Y'(t)+2Y(t)=X(t),Y(0)=...
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