Baum--Katz laws for certain weighted sums of independent and identically distributed random variables

The tail behaviour of partial sums of independent and identically distributed (i.i.d.) random variables X, X1, X2, ... depends on moment conditions on X. Whereas if the moment generating function exists in a neighbourhood of zero there are large-deviation principles giving precise information on the tail behaviour, at the other end of the scale, namely if only lower-order moments exist, we have less precise information on the tails of the distribution function of Sn ,= E'Xj, which can be expressed in the so-called Baum-Katz laws. These theorems reflect exactly the moments available. Starting with papers by Erdds, Katz, and Baum and Katz, various results have evolved. We formulate one version; see, for example, Baum and Katz (1965). Throughout our paper X, Xk, k E N, denote i.i.d. random variables.