I want to prove the following weak law of large numbers, similarly used in Rogers "Arbitrage with fractional Browian Motion" (1997): Assuming $(X_t)_{t \geq 0}$ is a fractional Brownian motion of Hurst parameter $0<H<1$ and let $p>0$ be a constant. I want to show that $$ Y_{n,p} := \frac{1}{n} \sum_{t=1}^n |X_t -X_{t-1}|^p \, \overset{P}{\longrightarrow} \, \mathbb{E}|X_1 -X_0|^p \quad, \text{as } n \to \infty $$ where $\overset{P}{\longrightarrow}$ refers to convergence in probability. I have found a source which proofed that it is sufficient to show that the autocovariance converges to zero, i.e. $$ \gamma(h) = \text{Cov}(|X_{t+h} -X_{t+h-1}|^p, |X_t -X_{t-1}|^p) \overset{h \to \infty}{\longrightarrow} 0 $$ (because if we know that $|X_t -X_{t-1}|^p$ is a weakly stationary process, this implies that $\text{Var}(Y_{n,p}) \to 0$ as $n \to \infty$ and we would get convergence in probability by the Chebychef inequality).
I was already able to show that $$ \widetilde{\gamma}(h) = \text{Cov}(X_{t+h} -X_{t+h-1}, X_t -X_{t-1}) \overset{h \to \infty}{\longrightarrow} 0 $$ but I dont really now where to go from here on forward. I was hoping that I could use the convergence of $\widetilde{\gamma}(h)$ for the convergence of $\gamma(h)$ but I don't really now how?
I know that there exist some ergodic theorems that even imply almost-sure convergence, but I am not familiar with ergodic theory at all and was wondering if I could use the above approach.
Thank you very much for taking the time to consider, I would really appreciate it if someone could help me out or give me a hint.