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In this paper non-asymptotic exponential and moment estimates are derived fortail of distribution for discrete time martingale and martingale transform bymeans of martingale differences in the terms of moments and tails ofdistributions of summands and multipliers.  We show also the exactness of obtained estimations and consider someapplications in the theory of limit theorem for Banach space valuedmartingales.

Moment and tail estimates for martingales and martingale transform, with application to the martingale limit theorem in Banach spaces