On the distribution of maxima of martingales

In the present paper we prove the boundedness of second exponential moments for the suprema of norms of partial sums of random series in Banach spaces, Sidon lacunary series on a compact Abelian group with coefficients in a Banach space, and for random lacunary series with similar coefficients. Exponential moments related to the law of iterated logarithm for lacunary series are also proved to be finite. Our results, even in case of standard trigonometric series with real coefficients, strengthen the classical theorem of Salem and Zygmund.