On the convergence of moments in the almost sure central limit theorem for martingales with statistical applications

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A martingale approach for the elephant random walk (2017, unknown): “in Theorem 3 of [3], lim n→∞ 1 log vn n”
On the Multi-dimensional Elephant Random Walk (2017, background): “14) that lim n→∞ nfn = 1−2a where fn = a2 n vn , which of course implies that fn converges to zero. Therefore, it follows from the quadratic strong law for real martingales given e.g. in Theorem 3 of [4], that for any vector u of Rd, (B.12) lim n→∞ 1 logvn Xn k=1 fk M2 k(u) vk = 1 d kuk2 a.s. 12 BE”