By William F. Stout

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**Extra info for Almost Sure Convergence**

**Example text**

Let 0 = x (0n) < x ^ < • • • < x4n> = 1 define a set of increasingly finer partitions becoming dense in [0, 1]. Let /( • ) be a measurable real valued function of bounded variation defined on [0, 1]. Let A[n) = [0, x<">] and A\n) = (xlul, x-n)] for 2 < / < n and each n > 1. 34 Chapter 2 POINTWISE CO N VERG EN CE OF PARTIAL SUMS For each « > 1 and 0 < x < 1 let i when i —1 xzA? for some 1 < / < n. For each n > 1, let be the a field generated by unions of the A\n\ Let Q = [0,1], & be the Borel sets of [0,1], and P be a Lebesgue measure.

3. 1. Fix an integer n > 1. Under the assumption of infinite capital for both the gambler and the house, devise a strategy for the gambler that guarantees that he goes home n dollars richer. Devise a strategy such that his expected take home winnings are infinite. | The martingale assumption is rather weak. Thus there are many and diverse examples of martingales and as a result martingale theory is widely applied, extending far beyond the modeling of fair games. We mention several examples to give some feeling for this diversity.

6 1 = * 0 ,3 2 + * 3 2 ,4 8 + * 4 8 ,5 6 + * 5 6 ,6 0 + * 6 0 ,6 1 - Thus each expansion then has at most k + 1 = 7 terms in it. ) Represent the expansion of Si by Si = t (h - 3=1 1) - Then Si2 < 7 £ follows by the Cauchy-Schwarz inequality. Now Z * ^ - l . ^ — * 0 ,6 4 + ( * 0 ,3 2 + * 3 2 ,6 4 ) + ( * 0 ,1 6 + * 1 6 ,3 2 + * 3 2 ,4 8 + * 48,64) + *• • + (*0,1 + * 1% + *** + * 63,64)Thus (the key step) max S f < 1[X02m + (*o%2 + * 322,64) + (*0,16 + * 16,32 1<2<26 + * 3 2 ,4 8 + * 4 8 ,6 4 ) + **’ + (* 0 2,1 + * 1 ,2 + *#* + * 6 3 ,6 4 ) 1 - There are k + 1 = 7 parenthetical expressions inside the square brackets.