= Solution
For a maximum of finitely many nonnegative quantities, its $p$th power is at most the sum of their $p$th powers. Hence the increment assumption gives
$$
\mathbb EA_n^p\leq\sum_{k=0}^{2^n-1}\mathbb E|X_{(k+1)2^{-n}}-X_{k2^{-n}}|^p
\leq C^p2^{n(1-p\beta)},\qquad
\|A_n\|_p\leq C2^{-n(\beta-1/p)}.
$$
By the <triangle inequality> in the <Lp norm>, the partial sums defining $K_\alpha$ have norms bounded by a geometric series. <Monotone convergence> of their $p$th powers gives
$$
\boxed{\|K_\alpha\|_p\leq\frac{2C}{1-2^{-(\beta-1/p-\alpha)}}<\infty.}
$$
In particular $K_\alpha$ is finite <almost surely>, so the preceding pathwise estimate has a common probability-one domain for all dyadic pairs.
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