Does a power spectral density that follow a power law imply that the time hisory follows a power law?

This is a question related to this previous question "I cannot find reference (paper) of this relation $$u(t)t^{α} ↔^{FT} f^{-(α+1)}$$".

The signal that I work with g(t), is the time history of a velocity component flucutuations. This signal comply with the rule:

$$\bar{g}=lim_{T->+\infty}\int_{0}^{T}g(t)dt=0$$

The Power spectrum of g(t) follows in a frequency range a power law "-5/3", in other frequency ranges the slope can change. (SEE for example, figure 1 [here])2. The power law frequency range, following the rule $$u(t)t^{α} ↔^{FT} f^{-(α+1)}$$ should imply that also the time history has a power law. The signals I work with does not seems to have a power law. Is this can be due to a masking performed by the behaviour of the other frequency range? The rule

$$\bar{g}=lim_{T->+\infty}\int_{0}^{T}g(t)dt=0$$

can violate some hypotheses at the base of $$u(t)t^{α} ↔^{FT} f^{-(α+1)}$$" ?

In your case $$u(t) \cdot t^{\alpha}$$ is one signal that fits your PSD but there are many others that will fit too but have a totally different shape.
This being said, your observation of a conflict has some merit. Your first equation simply states that $$g(t)$$ is mean free which in turns implies that the PSD at $$f=0$$ must be zero, but $$0^{-5/3}=1$$. I'm guessing that your power law applies only to a limited frequency range which doesn't include $$f=0$$ and hence there is no conflict.