The author defines the following characteristic for a random variable \(\xi\) with values in a Banach space X with the norm \(\| \cdot \|:\) \(q(\epsilon,\xi)=\inf_{f\in X'}{\mathbb{P}}(f(\xi)\geq -\epsilon)\), where X’ is the unit ball in the dual space \(X^*\). The main aim of this paper is to prove the following theorem: If \(q(\epsilon,S_ n-S_ k)\geq q\) for \(k=1,...,n\) then \(q{\mathbb{P}}(Z_ n>t)\leq {\mathbb{P}}(\| S_ n\| >t- \epsilon)\) holds, where \(S_ k=\xi_ 1+...+\xi_ k\), \(Z_ n=\max_{1\leq k\leq n}\| S_ n\|\). Three other theorems in connection with generalizations of the Lévy inequality are also proved.