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Characteristic function

  1. Jul 19, 2009 #1
    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. jcsd
  3. Jul 19, 2009 #2
    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
     
  4. Jul 19, 2009 #3
    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.
     
  5. Aug 8, 2009 #4
    A good book on characteristic functions is by Lukacs (called characteristic functions).

    In your example if you have [itex]v(x,t)=y(t)z(x)[/itex], then the cf would be
    [tex]\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[/tex]
     
  6. Aug 10, 2009 #5
    thanks a lot for the information. that reference will be usefull.
     
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