- #1
mikhairu
- 2
- 0
Hello,
I am trying to find a characteristic function (CF) of a Compound Poisson Process (CPP) and I am stuck :(.
I have a CPP defined as X(t) = SIGMA[from j=1 to Nt]{Yj}. Yj's are independent and are Normally distributed.
So, in trying to find the CF of X I do the following:
(Notation: CFy = Characteristic Function of y (y is subscript))
CF(X) = E[exp{i*u*X}]
= E[ (E[exp{i*u*Y}])^N ]
= E[ (CFy(u))^N ]
= E[ (exp{ ln[CFy(u)] })^N ]
which is really just a moment generating function Phi_N (Phi subscript N):
Phi_N( ln[CFy(u)] ).
I don't know how to go from here... Y's are Normally distributed but the entire process is Poisson.. so I'm not sure how to combine these. Please help!
Thank you.
I am trying to find a characteristic function (CF) of a Compound Poisson Process (CPP) and I am stuck :(.
I have a CPP defined as X(t) = SIGMA[from j=1 to Nt]{Yj}. Yj's are independent and are Normally distributed.
So, in trying to find the CF of X I do the following:
(Notation: CFy = Characteristic Function of y (y is subscript))
CF(X) = E[exp{i*u*X}]
= E[ (E[exp{i*u*Y}])^N ]
= E[ (CFy(u))^N ]
= E[ (exp{ ln[CFy(u)] })^N ]
which is really just a moment generating function Phi_N (Phi subscript N):
Phi_N( ln[CFy(u)] ).
I don't know how to go from here... Y's are Normally distributed but the entire process is Poisson.. so I'm not sure how to combine these. Please help!
Thank you.