 Solution

Write $A_n={\max }_{0\le k<2^n}|{\xi }_{(k+1)2^{-n}}-{\xi }_{k{2}^{-n}}|$. The maximum of nonnegative numbers to the power $p$ is bounded by their sum, so the increment hypothesis givesTherefore $\Vert A_n{\Vert }_p\le C{2}^{-n(\beta -1/p)}$. The Minkowski inequality and completeness of $L^p$ imply convergence of the defining series wheneverFor a fully explicit argument, the norm of every tail is at most $2C{\sum }_{n\ge N}{2}^{-n(\beta -1/p-\alpha )}$, which tends to zero. Since the summands are nonnegative, their pointwise increasing sum equals this finite $L^p$ limit and is finite almost surely. This is a guaranteed range; no endpoint convergence follows from the displayed summability estimate. The assumptions control increments only and do not require ${\xi }_0\in L^p$.