Page 82 - Elements of Distribution Theory

P1: JZP
052184472Xc02 CUNY148/Severini May 24, 2005 2:29

68 Conditional Distributions and Expectation

and Woodroofe (1975, Chapter 10) give more elementary, but still very useful, discussions of condi-
tioning.
Exercise 2.19 briefly considers an approach to conditional expectation based on projections in
spaces of square-integrable random variables. This approach is developed more fully in Karr (1993,
Chapter 8).
Exchangeable random variables are considered in Port (1994, Chapter 15). Schervish (1995,
Chapter 1) discusses the relevance of this concept to statistical inference.
Martingales play an important role in probability and statistics. The definition of a martingale
used in this chapter is a special case of a more general, and more useful, definition. Let X 1 , X 2 ,...
and Y 1 , Y 2 ,... denote sequences of random variables and suppose that, for each n = 1, 2,..., X n
is a function of Y 1 , Y 2 ,..., Y n . The sequence (X 1 , X 2 ,...)is said to be a martingale with respect to
(Y 1 , Y 2 ,...)if

E(X n+1 |Y 1 ,..., Y n ) = X n , n = 1, 2,....
Thus, the definition used in this chapter is a special case in which Y n is taken to be X n , n = 1, 2,....
See, for example, Billingsley (1995, Section 35), Karr (1993, Chapter 9), Port (1994, Chapter 17),
and Woodroofe (1975, Chapter 12) for further discussion of martingales.