Probability inequalities for sums of independent random variables with values in a Banach space

where x > 0, y > O,A EIXI < ,t > 2andK 1 + e-t(t + 1)t+2. This paper is devoted to improving this result and to extending it to the case of non-identically distributed independent random variables for which the existence of finite moments ofsome particular order is not assumed. In Section 1, certain inequalities are derived whose right-hand sides consist of two components the sum of the probabilities of the tails and a component containing truncated moments. In Section 2, the proofs of these inequalities are given. A bilateral inequality is stated in Section 3. Special cases are considered in Section 4. Section 5 deals with examples involving the computation of the probabilities and Section 6 contains applications to the strong law of large numbers. Let X, ..., X, be non-identically distributed independent random variables (i.r.v.) with respective distribution functions F(u),..., F,(u). Set