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I'm reading a tutorial on stochastic processes. There is an example in the tutorial as follows:

General Moving Average random process given as $X[n]=\frac{(U[n]+U[n-1])}{2}$ where $E[U[n]]=\mu$ and $var(U[n]) = {σ^2}_U$ and the $U[n]$'s are uncorrelated.

As you see $X[n]$ is a 1-D random variable Then the example is solved in the following way:

$[C_X]_{ij}=E[(X[i]-E[X[i]])(X[j]-E[X[j]])]\qquad i=0,1,\dots,N-1;j=0,1,\dots,N-1.$

$\begin{align} X[n]-E[X[n]]&=\frac{1}{2}(U[n]+U[n-1])-\frac{1}{2}(\mu+\mu)\\ &=\frac{1}{2}[(U[n]-\mu)+(U[n-1]-\mu)]\\ &=\frac{1}{2}[\overline U[n]+\overline U[n-1]] \end{align}$

$\begin{align} [C_X]_{ij}&=\frac{1}{4}E[(\overline U[i]+\overline U[i-1])(\overline U[j]+\overline U[j-1])]\\ &=\frac{1}{4}(E[\overline U[i]\overline U[j]]+E[\overline U[i]\overline U[j-1]]+E[\overline U[i-1]\overline U[j]]+E[\overline U[i-1]\overline U[j-1]]) \end{align}$

$[C_X]_{ij}=\frac{1}{4}(\sigma^2_U\delta[j-i]+\sigma^2_U\delta[j-i-1]+\sigma^2_U\delta[j-i+1]+\sigma^2_U\delta[j-i]).$

$C_X=\begin{bmatrix}\frac{\sigma^2_U}{2}&\frac{\sigma^2_U}{2}&0&0&\dots & 0&0&0\\ \frac{\sigma^2_U}{4}&\frac{\sigma^2_U}{2}&\frac{\sigma^2_U}{4}&0&\dots &0&0&0\\ \vdots &\vdots &\vdots& \vdots&\vdots &\vdots &\vdots& \vdots\\ 0&0&0&0&\cdots &\frac{\sigma^2_U}{4}&\frac{\sigma^2_U}{2}&\frac{\sigma^2_U}{4}\\ 0&0&0&0&\cdots &0&\frac{\sigma^2_U}{4}&\frac{\sigma^2_U}{2}\end{bmatrix}$

So why $C_X$ is $n\times n$ inspite of $X[n]$ being 1-dimensional?

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  • $\begingroup$ I'm not sure I understand. If you have $n$ 1D random variables, their covariance matrix is $n \times n$, in that the $(i,j)$ entry is the covariance between the $i$th variable and the $j$th variable. $\endgroup$
    – Ian
    Aug 3, 2015 at 22:18
  • $\begingroup$ I mean X[n] is just one 1-D random variable that is a function of n 1-D random variables. But I'm not sure!!!! $\endgroup$ Aug 3, 2015 at 22:20
  • $\begingroup$ It looks to me like your covariance matrix is that of $X[1],X[2],\dots,X[n]$. So you have $n$ 1D variables. $\endgroup$
    – Ian
    Aug 3, 2015 at 22:23
  • $\begingroup$ @Augustin can you please solve my misunderstanding here? $\endgroup$ Aug 4, 2015 at 3:05

1 Answer 1

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First, your random process $X[n], n \in \mathbb{N}$ is collection of infinitely many random variables $X[1],X[2],\ldots$; if we take finitely many of them, say, $X[1],\ldots,X[n]$ and form random vector $$X=(X[1],\ldots,X[n])$$ this random vector is sometimes called a finite-dimensional section of random process $X[n], n \in \mathbb{N}$.

With every random vector $Y=(Y_1,\ldots,Y_n)$ we can associate its covariance matrix $C_Y=\{c_{ij}\}_{i,j \in \{1,..,n\}}$ with entries $$c_{ij}=Cov(Y_i,Y_j)=E[(Y_i-E[Y_i])(Y_j-E[Y_j])].$$ In your tutorial, $C_X$ is covariance matrix of finite-dimensional section $X$ of random process $X[n], n \in \mathbb{N}$ introduced in the previous paragraph.

When $Y$ is not a random vector, but a proper stochastic process, in place of covariance matrix we use auto-covariance function $$C_Y(t_1,t_2)=Cov(Y_{t_1},Y_{t_2})=E[(Y_{t_1}-E[Y_{t_1}])(Y_{t_2}-E[Y_{t_2}])]$$ Of course, when process $Y$ is WSS, auto-covariance function $C_Y(t_1,t_2)$ depends on $(t_1,t_2)$ only through the lag $h=t_2-t_1$, and we can write auto-covariance function as $$C_Y(h)=Cov(Y_0,Y_h).$$

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  • $\begingroup$ please look at the question sufficient conditions for a stochastic process to be wide sense stationary and this chat between me and augustin where we say $X(t) = B + A \sin(\omega_0 t + \Phi$ is a 1-D random variable for each $t$ and so it's covariance and correlation matrices should be $1\times 1$. here I mean $X[n]$ is a 1-D random variable for each $n$ and so its covariance and $\endgroup$ Aug 4, 2015 at 14:10
  • $\begingroup$ correlation matrices should be $1\times 1$. If I'm wrong we can discuss in chat because I'm really confused now $\endgroup$ Aug 4, 2015 at 14:12
  • $\begingroup$ @sepideh OK, I am available for chat in in the next few hours, if you are still interested. $\endgroup$ Aug 4, 2015 at 15:12

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