Suppose a1, a2,... is a sequence of real numbers with an → ∞. If $\lim \sup(X_1 + \cdots + X_n)/a_n = \alpha$ a.s. for every sequence of independent nonnegative uniformly bounded random variables X1, X2,... satisfying some hypothesis condition A, then for every (arbitrarily-dependent) sequence of nonnegative uniformly bounded random variables $Y_1, Y_2,\ldots, \lim \sup(Y_1 + \cdots + Y_n)/a_n = \alpha$ a.s. on the set where the conditional distributions (given the past) satisfy precisely the same condition A. If, in addition, $\Sigma^\infty a^{-2}_n < \infty$, then the assumption of nonnegativity may be dropped.
This monthly journal, begun in 1950, is devoted entirely to research in pure and applied mathematics, principally to the publication of original papers of moderate length. A section called Shorter Notes was established to publish very short papers of unusually elegant and polished character for which there is normally no other outlet.
Founded in 1888, to further mathematical research and scholarship, the 30,000-member American Mathematical Society provides programs and services that promote mathematical research and its uses, strengthen mathematical education, and foster awareness and appreciation of mathematics and its connections to other disciplines and everyday life. The headquarters of the AMS are in Providence, Rhode Island. The Society also maintains a government relations office in Washington, D.C., the Mathematical Reviews editorial office in Ann Arbor, Michigan, and a warehouse and distribution facility in Pawtucket, Rhode Island. The Society has approximately 240 employees.
This item is part of a JSTOR Collection.
For terms and use, please refer to our
Proceedings of the American Mathematical Society
© 1983 American Mathematical Society