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Let X e L'(N, F,P) and let G,H be sub o-algebras of F. Moreover be independent of o(o(X),G). Show that E{X|o(G, H)} = E{X\G}.

Let X e L'(N, F,P) and let G,H be sub o-algebras of F. Moreover be independent of o(o(X),G). Show that E{X|o(G, H)} = E{X\G}.

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
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23.16 Let X E L'(N, F, P) and let G, H be sub o-algebras of F. Moreover let H be independent of o(o(X), G). Show that E{X\o(G,H)} = E{X\G}. 23.17 Let (Xn)n>1 _be independent and identically distributed and in L' and let Sn E-1 X; and Gn = 0(Sn, Sn+1;,...). Show that E{X1\Gn} = E{X1 | Sn} and also E{X;|Gn} = E{X; | Sn} for1<j<n. Also show that E{X;\Gn} = E{X1|Sn} for 1 < j <n (Hint: Use Exercise 23.16.) 23.18 Let X1, X2,. .., Xn be independent and identically distributed and in L1. Show that for each j, 1 < j < n, we have E{X;|E, Xi} = ¿£, X;. (Hint: Use Theorem 23.2 and the symmetry coming from the i.i.d. hypothe- sis.) i=1
Transcribed Image Text:23.16 Let X E L'(N, F, P) and let G, H be sub o-algebras of F. Moreover let H be independent of o(o(X), G). Show that E{X\o(G,H)} = E{X\G}. 23.17 Let (Xn)n>1 _be independent and identically distributed and in L' and let Sn E-1 X; and Gn = 0(Sn, Sn+1;,...). Show that E{X1\Gn} = E{X1 | Sn} and also E{X;|Gn} = E{X; | Sn} for1<j<n. Also show that E{X;\Gn} = E{X1|Sn} for 1 < j <n (Hint: Use Exercise 23.16.) 23.18 Let X1, X2,. .., Xn be independent and identically distributed and in L1. Show that for each j, 1 < j < n, we have E{X;|E, Xi} = ¿£, X;. (Hint: Use Theorem 23.2 and the symmetry coming from the i.i.d. hypothe- sis.) i=1
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