Stochastic Integrals and Cauchy Variables

Question

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!

Answer 1

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 is a standard reference on general stochastic integration, which includes discussions of Lévy processes.

Answer 2

The Itô integral is appropriate when you are integrating against white noise $\mathrm dW_t$, where $W_t$ is a Brownian motion. I don't quite know how to handle your sum of Cauchy random variables, but I suspect the answer lies in integrating $f(t)$ against a colored noise $\mathrm dX_t$, which is characterized by a power-law tail decay instead of Gaussian tail decay. To have a proper stochastic integral formulation, we must find some process $X_t$ which represents the integral of colored noise.

The correct process to use is the Ornstein-Uhlenbeck process $X_t$, which is determined by two parameters: its variance $\sigma$ and correlation time $\tau$. The process $X_t$ solves the stochastic differential equation $$\mathrm d X_t = -\tfrac{1}{\tau} X_t \mathrm d t + \sqrt{\tfrac{2\sigma^2}{\tau}} \mathrm d W_t,$$ where $W_t$ is a Brownian motion.

I think the following reference may be useful for you:

• Colored Noise in Dynamical Systems by Hänggi and Jung

Answer 3

The sum of independent normal variables is normal and you can easily get the two parameters.

The same is true for independent Cauchy variables (they are Cauchy).

But apparently you are not looking for results about a given partial sum but for the trajectory...