# maximal inequality

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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## 1 Answer

Try On some maximal inequalities for fractional Brownian motions by Alexander Novikov and Esko Valkeila, Statistics & Probability Letters, 1999, vol. 44, issue 1, pages 47-54.

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Let $(B^{H}_{t})_{{t}\geq 0}$ be a fBM. Using stationarity we have $B^{H}_{t+r}-B^{H}_{t} = B^{H}_{r}$ in law. Then i could apply Theorem 1.1, right? –  Peter Moor Sep 3 '12 at 18:38
And am I right, assuming, that this holds for all $H\in\left(0,1\right)$?Because Iam not considering stopping times. –  Peter Moor Sep 3 '12 at 19:41
Can anyone give a feedback? –  Peter Moor Sep 10 '12 at 11:22