If I know the characteristic function phi_X(t) of a random variable X>0, how can I write the characteristic function phi_Y(t) of Y=log(X)?
Elisabeth Wiley
Answered question
2022-08-12
Characteristic function of logarithm of random variable If I know the characteristic function of a random variable , how can I write the characteristic function of ? I know that and . But I can't derive one from the other. Any idea? I would like to use to calculate the second moments
Answer & Explanation
Kyle George
Beginner2022-08-13Added 22 answers
This question comes up in various guises: knowing the Fourier transform of (in your case, the probability density function of ), can we find the Fourier transform of another related function (in your case, the p.d.f. of )? Unfortunately, nonlinear transformations completely mess up the picture of the Fourier transform. All one can do is to invert the Fourier transform, apply the desired nonlinear transformation, and take the Fourier transform of that.
Gauge Roach
Beginner2022-08-14Added 3 answers
We need to compute the log-characteristic function that is the characteristic of . So, we need to express the characteristic function of in terms of values :
As , we have and so :
We need to express the density of in terms of the density of . As , because the probabilities are invariant by change of variables, the transformation is strictly monotone and derivable in all the domain of X so that:
In the end, the log characteristic function equals: