The Law Of The Iterated Logarithm And The Strong Law Of Large Munbers For Product Sums Of PA Sequences

The definition of PA sequence, an important kind sequence of -dependent random variables, was introduced by Esary, Proschan & Walkup(1 967)(see definition 1.2). It not only properly includes the sequence of independent random variables, but also has wide applications in multivariate statistical analysis, reliability theory, percolation theory and other applied fields (see ref.[14]). Thus, the studies on the limit theory of PA sequences have received more and more attentions (see ?2). But up to now, the studies are only for partial sums and, haven抰 shown any concern on the product sums, however, the partial sums and the product sums not only have the osculating aspects, but also have essential difference between them (see example 1). So, the studies for product sums of PA sequences play an important role in theoretical and applied setups. By using the method of turning the product sums to the sums of product of partial sums in ref. [18], this paper proves the law of the iterated logarithm for product sums of strong stability sequence of PA sequences with different distributions, not only extend the corresponding results in Yu (1986), but also extend and improve the corresponding results in Birkel( 1989).