How Do You Solve Complex Fourier Integrals in Probability Theory?

[capitano_nemo]

Hi to everybody

I have to solve this Fourier integral:

1) f(q)=\int_{-infty}^{+infty}a*b*x^(b-1)*exp(-a*x^b)*exp(i*q*x)*dx

and if S_n=x_1+...+x_n, with S_n the sum of n random variables IID, then I can write:

f_n(q)=[f(q)]^n,(convolution theorem), then the anti-trasform of f_n(q) give the pdf of the variable S_n.

2) F(S_n)=(1/2*pi)*\int_{-infty}^{+infty}f_n(q)*exp(-i*q*x)*dq.

I must to solve the equations 1) and 2) in order to solve my problem, the equation 2) is the final solution of the problem.

Thanks

 

[suyver]

$$f(q)=ab\int_{-\infty}^{\infty}x^{b-1}\exp(-a\,x^b) \exp(i \,q\,x)dx$$

I don't think that this integral has an analytical solution...