Your proof below extends to all probability distributions.
I suggest you study this. General measure theorentic probability is not really more difficult than densities, and likelihood ratios from densities can have any distribution on the positive reals for which the expectation (integral as you use it) is 1.
> The first and last terms are both less than int[-inf,inf](x^2*f(x)dx) which is finite by hypothesis. The second therm is easily bounded:
> So mu is finite, i.e., 2. holds. Since sigma^2 = E[X_1^2] - mu^2, also 3. holds.
> Does it seem correct to you? I have no idea how to extend it to real random variables which are not continuous, though. Some help here? Thanks,
> Best Regards
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