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Let $(W_{t})_{t\geq 0}$ be a standard Brownian Motion. Then it is easy to get the following inequality, using the Burkholder inequality: $\mathbb{E}\left[sup_{r\in\left[0,s\right]}\left|W_{t+r}-W_{t}\right|^{q}\right]\leq C_{q}s^{\frac{q}{2}}$. Is it possible to get a similar result for the fractional Brownian Motion instead of the usual Brownian Motion? I have already found some Burkholder Inequalities for the fBM but I cant really apply them to get such a result

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