On probablity and moment inequalties for dependent random variables

The paper obtains the upper estimate for the probability that a norm of a sum of dependent random variables with values in the Banach space exceeds a given level. This estimate is principally different from the probability inequalities for sums of dependent random variables known up to now both by form and method of proof. It contains only one of the countable number of mixing coefficients. Due to the introduction of a quantile the estimate does not contain moments. The constants in the estimate are calculated explicitly. As in the case of independent summands, the moment inequalities are derived with the help of the estimate obtained.