By William F. Stout

ISBN-10: 0126727503

ISBN-13: 9780126727500

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**Example text**

It is interesting to note the connection between { R i ,i > 0 } and the binary expansion of numbers in the interval [0, 1]. Let Bi = (1 - Ri)/2 for i > 0. Then it is easily seen that for each co e [0,1], • • is the binary expansion of a>. Thus results about {R i, / > 0} can be interpreted as results about binary expansions of real numbers. 1. 54 POINTWISE CO N VERGEN CE OF PARTIAL SUMS Chapter 2 Our purpose for introducing the Rademacher system is the construction of the associated Haar system {Xiy i > 1}.

The Polya urn scheme). Let an urn contain b black and r red balls. Let T0 = b/(b + r). At each drawing a ball is drawn at random, its color is noted and a balls of that color are added to the urn. Let b„ be the number of black balls and an be the number of red balls after the nth drawing. 4. 5 35 MARTINGALE DIFFERENCE SEQUENCES the Hth drawing. E[Tn | T,n —1 9 ♦ • • >^i] — E\Tn I T’n -il 71—1 a*s* ^71-1 “t“ rn- 1 - Thus {Tn, n > 1} is a martingale. | (Likelihood ratio). Let {T£, / > 1} be a stochastic sequence for which the joint density of {Yl9 Y2, .

S. If 0 < cor(A", Y) < 1, then there exists a line of strictly positive slope such that (X, Y) has a joint distribution which tends to concentrate its mass around the line. The closer cor(Z, Y) is to 1^ the more pronounced is this concentration of mass. If —1 < c o r ^ , Y) < 0, the same remarks apply to a line with strictly negative slope. If cor(X, Y) = 0, such as when X and Y are independent, then there does not exist a line with nonzero slope having the above property. These remarks perhaps help in interpreting the following corollary.

### Almost Sure Convergence by William F. Stout

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