Characteristic function of a random variable X is defined as E[exp(itx)]. It has been studied a lot and a number of its useful properties are known. Is there known any "generalization" of characteristic function, e.g. of form E[exp(it f(x))], with a reasonably well developed similar "theory"?
If you mean E[exp(it f(X))] (upper case X instead of lowercase x), I am not sure how this would be a generalization as f(X) is just another random variable e.g. if X is normally distributed and f(x)=exp(x), then f(X) is log-normally distributed random variable.
If you mean E[exp(it f(X))] (upper case X instead of lowercase x), I am not sure how this would be a generalization as f(X) is just another random variable e.g. if X is normally distributed and f(x)=exp(x), then f(X) is log-normally distributed random variable.
I agree with Sebastian and do not think that a generalization is necessary. In fact, we can consider f(x) as a new random variable y, specify the distribution of y via the destribution of x by the standard transformations, and that is it.
Thank you for your answers. You are right! Of course, I have missed the obvious thing...
May be you can look at Fourier Transformation (FT) and it generalizations as FT is closely related to characteristic function.
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