No abstract

No abstract

Y1,Y2,.. . be a sequence of independent two-valued random variables, Yn+i =-s or f3 + ns with probabilities (1-W) and W, where s is a small positive number and W is then determined by the condition-E(Y.) = aV(Yn). Verify that the probability that for some n,-3 + Y, +. .. + Yn > 0 converges to 1/(1 + ad) as s-. 0, and let X,, = Yn-E(Yn). This completes the proof. The theorem can be extended to say that for each y > 0, if rT is the least n if any for which (Xi +. .. + Xn) .-yT +GI +. .. + An) + .(V. + * * * + Vn). then the probability that there is some n < rz for which (1) holds is less than (,y/((+ 3))(1/(1 + ac)); and this bound is sharp. The material of this note, including proofs of Lemmas 1 and 2, will appear as part of our forthcoming book, How to Gamble If You Must (New York: McGraw-Hill), Theorems 2.12.1 and 9.4.1, and an illustrative application of the theorem will appear in the forthcoming article, "A sharper form of the Borel-Cantelli lemma and the strong law" by L. E. Dubins and D. A. Freedman.

No abstract

No abstract