Characteristic function

1. Jul 19, 2009

emptyset

Hi.
Does anyone know a good source for learning about the characteristic function of a joint pdf. Is there any nice rules for that? For example assume having a waiting time density and a jump density which are independent (easy things first). Is there an elegant way to get the characteristic function of the process if I have the two individual cf's?
Thanks

2. Jul 19, 2009

John Creighto

According to wikipedia for independent random variables the joint characteristic function is just the product of the characteristic function for each random variable.

http://en.wikipedia.org/wiki/Charac..._theory)#Basic_manipulations_of_distributions

3. Jul 19, 2009

emptyset

Thanks for your reply. This is true for a sequence of independent (and not necessarily identically distributed) random variables. However, I am looking for the joint pdf.
As an example, assume you have a partical sitting around at x' for a random time tau with the distribution y(t) and then jumping to x'' with the jump distribution z(x).
I am looking for a nice way of dealing with the characteristic funtion of the joint pdf v(x,t)=y(t)z(x)
given I now the charactersitic functions of y and z.

4. Aug 8, 2009

Focus

A good book on characteristic functions is by Lukacs (called characteristic functions).

In your example if you have $v(x,t)=y(t)z(x)$, then the cf would be
$$\psi(\theta)=\int_{\mathbb{R}}\int_{\mathbb{R}}e^{i\theta x t}z(x)y(t) dx dt =\int_{\mathbb{R}}\psi_z(t\theta)y(t) dt$$

5. Aug 10, 2009

emptyset

thanks a lot for the information. that reference will be usefull.