I hope there is a straighforward literature-pointer here.


If I were interested in $\sum\_{t=1}^{n} f(t) X\_{t}$, where $X\_{t}$ consists of independent normal random variables, I could approximate the sum as an Ito integral, and then (if $f(t)$ is reasonably nice) get a good answer for the resulting approximation. Also, my impression is that this is really the 'best approach' as long as $n$ is getting big and $f(t)$ isn't too wildly spiky.


Is there an analogous theory when $X\_{t}$ is Cauchy? 


I'm aware that there are lots of 'infinity issues' around adding up Cauchy variables, e.g. that sums with equal weights are dominated by their biggest term and so on... but I'm still hoping that there is a somewhat unified approach for looking at this type of problem.


Thanks!



Stochastic Integrals and Cauchy Variables
Let $\{Y(t):t\ge 0\}$ be a symmetric 1-stable Lévy process. Then $Y$ is a càdlàg process with $E[e^{iuY(t)}]=e^{-t|u|}$. A Levy process is a semimartingale, so we may define the usual stochastic integral with respect to $Y$. Consequently, if $f$ is continuous, then the stochastic integral is the limit in probability of left-endpoint Riemann sums. For example,
 $$
 \sum\_{j=1}^{n} f(t\_{j-1})(Y(t\_j) - Y(t\_{j-1}))
 \to \int\_0^1 f(s)\,dY(s),
 $$
in probability as $n\to\infty$, where $t\_j=j/n$.


Since a Lévy process has stationary, independent increments, the sum on the left is equal in law to
 $$
 \frac1n\sum\_{j=1}^{n} f(j/n)X\_j,
 $$
where $\{X\_j\}$ is an iid sequence of standard Cauchy random variables, i.e. $E[e^{iuX\_1}]=e^{-|u|}$ and $X\_1$ has density $1/\pi(1+x^2)$.


[Protter](http://books.google.com/books/about/Stochastic_integration_and_differential.html?id=mJkFuqwr5xgC) is a standard reference on general stochastic integration, which includes discussions of Lévy processes.


