On the duality between the behaviour of sums of independent random variables and the sums of their squares

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Abstract
Let X nj , 1 ≤ j ≤ k n , be independent, asymptotically negligible random variables for each n ≥ 1. In certain cases there exists a duality between the behaviour of Σ j X nj and . We extend one of the known forms of this duality, and show that, under mild conditions on the truncated moments of the X nj , the convergence of to 1 in the mean of order p (p ≥ 1) is equivalent to the convergence of Σ j X nj to the standard normal law, together with the convergence of its 2pth absolute moment to that of a standard normal variable. A similar result holds in the case of convergence to a Poisson law.