Note: I'm not an expert on stochastic processes. Please use small words and speak real slow.


I'm reading a paper [[1](http://projecteuclid.org/download/pdfview_1/euclid.aoas/1267453933 "[1]")], which uses a notation for stochastic processes that doesn't seem to be standard.



> 
> Now we generalize the squared covariance and define the square of
> conditional covariance, given two real-valued stochastic processes
> $U(·)$ and $V(·)$. We obtain an interesting result when $U$ and $V$ are
> independent Weiner processes. First, to center the random variable $X$
> in the conditional covariance, we need the following definition. Let $X$
> be a real-valued random variable and ${U (t):t ∈ ℝ}$ a real-valued
> stochastic process, independent of $X$. The $U$-centered version of $X$
> is defined by
> 
> 
> $$X\_U = U (X) − \int^∞\_{-∞} U (t) dFX(t) = U (X) − E[U (X)|U]$$
> 
> 
> 


In my experience a stochastic process is a collection of random variables $\{X\_t\}$... What is the intuition behind a stochastic process $U(·)$? Also, they have both $U(X)$ and $U(t)$ which I can't make heads or tails of.


Cheers



Stochastic Process Notation
$U(X)$ just means the stochastic process at a **random time** $X$. So you have two different random things, the stochastic process (collection of random variables) $\{U(t)\}$, and the random time $X$ picks out a particular one $U(X)$.


For instance, imagine you pick a random book from the shelf and open it on a random page. Then $\{U(t)\}$ is the book, $X$ is the page number, and $U(X)$ is the text on that page.


