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Abstract

Let $N\geq 2$ be a given integer. Suppose that $df=(df_n)_{n\geq 0}$ is a martingale difference sequence with values in $\ell_1^N$ and let $(\varepsilon_n)_{n\geq 0}$ be a deterministic sequence of signs. The paper contains the proof of the estimate $$ \mathbb{P}\Bigl(\sup_{n\geq 0}\, \Bigl\|\sum_{k=0}^n \varepsilon_k df_k\Bigr\|_{\ell_1^N}\geq 1\Bigr) \leq \frac{\ln N+\ln(3\ln N)}{1-(2\ln N)^{-1}}\sup_{n\geq 0}\mathbb E \Bigl\|\sum_{k=0}^n df_k\Bigr\|_{\ell_1^N}. $$ It is shown that this result is asymptotically sharp in the sense that the least constant $C_N$ in the above estimate satisfies $\lim_{N\to \infty}C_N/\!\ln N=1$. The novelty in the proof is the explicit verification of the $\zeta$-convexity of the space $\ell_1^N$.