Is there a Karhunen-Loeve theorem for discrete-time process?


For example, let $\left\{X\_i\right\}$ be a sequence of independent random variable which are uniformly distributed on the set $\{-1,1\}$. Define
$$
S\_t = \sum\_{i=1}^t X\_i,
$$
with $t$ an integer between 1 and $n$.
I would like to write $S$ as
$$
S\_t = \sum\_{j=1}^t \xi\_j e\_j \tag{1}
$$
where the coefficient $\xi\_j$ are random and pairwise independent.  

The vectors $\left\{e\_j\right\}\_{j=1}^n$ form an orthonormal basis of $\mathbb R^n$.  

The representation (1) should minimzed the total square error. 


I tried to adapt the proof of the Karhunen-Loeve and Mercer theorem.  

My guess is that the Mercer kernel, or covariance function, should now be
$$
K(s, t) = E\left(X\_s X\_t \right).
$$


I'm wondering if this kind of result is either: 


* already available from the standard theory? if so, how?  

(by appropriately defining the space maybe)
* already treated somewhere? if so where?  

(reproducing kernel Hilbert space theory, or a generalization of the Karhunen-Loeve expansion)


I seek the representation (1) because I want to discretize, or quantize, the process $S$.



Karhunen-Loeve expansion for discrete-time process
There does not seem to be a problem. I assume that you have a random field 


$$S\_t: \{1,\dotsc, n\}\to \mathbb{Z}$$


The covariance kernel is $(t\leq \tau)$ $\newcommand{\bE}{\mathbb{E}}$


$$ \bE(S\_tS\_\tau)=\bE(S\_t^2)= \sum\_{i=1}^t \bE(X\_i^2)=t=\min(t,\tau). $$


Now all you need to do is find the eigenvalues and eigenvectors of the $n\times n$ symmetric matrix $(\min(i,j))\_{1\leq i,j\leq n}$.


**Note:** Have a look at section 1.4 of [these notes by Zeitouni.](http://www.wisdom.weizmann.ac.il/~zeitouni/notesGauss.pdf)


