How Can You Determine the Characteristic Function of Joint PDFs?

[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

 

[John Creighto]

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

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

 

[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 particle 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.

 

[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
$$\psi(\theta)=\int_{\mathbb{R}}\int_{\mathbb{R}}e^{i\theta x t}z(x)y(t) dx dt =\int_{\mathbb{R}}\psi_z(t\theta)y(t) dt$$

 

[emptyset]

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