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Almost Sure Convergence by William F. Stout

By William F. Stout

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9 u n_l 9 u n9u n 9 .. 38 POINTWISE CO N VERGEN CE OF PARTIAL SUMS Chapter 2 Let = SS{U lf), i < «). Here and throughout the book denotes the a field generated by {Xx, a e A}. 1. Prove that {£/#>, n > 1} is a stochastic sequence. | It is now convenient to introduce a new notation. Throughout the book for a given A e ^ the indicator random variable 1(A) is defined by if coe A if co £ A. One of the most useful tools in establishing convergence results for mar­ tingales is the fact that optional stopping preserves the martingale structure.

8 APPLICATIONS OF BASIC CO N V ERG EN CE THEOREM 53 {Rt >i > 0}. Let *«>(«>) = { _ J if 0 < со < £ if \ < œ < 1. In order to define jR* for each i > 1, extend R0 periodically to [0, oo) and let Ri(a>) -- Roil'd)) for 0 < at < 1 and each i > 1. 1). As is easily shown, the R{ are independent identically distributed with P[Rt = 1] = P[Ri = — 1] = Thus {Riy i > 0} is merely one representation of a very familiar object, the “coin tossing” random sequence. It is interesting to note the connection between { R i ,i > 0 } and the binary expansion of numbers in the interval [0, 1].

A+1 X*)2 first, we motivate the choice of g. ( a+n r a+n \ 2 t X XA = E X X f + 2 X XJCA i= a + 1 / Li=a+1 a + l^ i< j< a + n J a+n < j; i= a + 1 a+n = £ i= a + l by Eq. 8). + 2 X e j - ^ x tE V 'x f a+l < i < j

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