On the Maximal Lévy-Ottaviani Inequality for Sums of Independent and Dependent Random Vectors Zbigniew S. Szewczak — 2013 Bulletin of the Polish Academy of Sciences. Mathematics We prove that the sums S k of independent random vectors satisfy P ( m a x 1 ≤ k ≤ n ∥ S k ∥ > 3 t ) ≤ 2 m a x 1 ≤ k ≤ n P ( ∥ S k ∥ > t ) , t ≥ 0.