= Solution
Write $A_n=\max_{0\leq k<2^n}|\xi_{(k+1)2^{-n}}-\xi_{k2^{-n}}|$. The maximum of nonnegative numbers to the power $p$ is bounded by their sum, so the increment hypothesis gives
$$
\|A_n\|_p^p\leq\sum_{k=0}^{2^n-1}\mathbb E|\xi_{(k+1)2^{-n}}-\xi_{k2^{-n}}|^p
\leq C^p2^n2^{-np\beta}.
$$
Therefore $\|A_n\|_p\leq C2^{-n(\beta-1/p)}$. The <Minkowski inequality> and completeness of $L^p$ imply convergence of the defining series whenever
$$
\boxed{0<\alpha<\beta-\frac1p,\qquad
\|K_\alpha\|_p\leq\frac{2C}{1-2^{-(\beta-1/p-\alpha)}}.}
$$
For a fully explicit argument, the norm of every tail is at most $2C\sum_{n\geq N}2^{-n(\beta-1/p-\alpha)}$, 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 $\xi_0\in L^p$.
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