Extension of the Brownian-motion moment theorem to real $\alpha$

Let $X^*$ be the process considered in Theorem, and let $\alpha$ be the parameter appearing in the regular-variation assumption. The theorem currently assumes that $\alpha$ is an integer. Conjecture. Theorem holds for any real $\alpha>1$. This would extend the Brownian-motion case beyond the integer values of the regular-variation parameter; the source gives no resolution.