CYTOPLASMIC INHERITANCE OF THE ORGANIZATION OF THE CELL CORTEX IN PARAMECIUM AURELIA

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: McGrawHill), 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.