This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Stat 219  Stochastic Processes Homework Set 3, Autumn 2009 1. Exercise 1.4.31 . Prove Proposition 1.4.3 using the following steps. (a) Verify that the identity (1.4.1) holds for indicator functions g ( x ) = I B ( x ) for B B . ANS: Let B B be an arbitrary Borel set and let g ( x ) = I B ( x ). Note that I B ( X ) = I { X B } . Hence, E ( g ( X )) = E ( I B ( X )) = E ( I { X B } ) = P X ( B ) = Z IR I B ( x ) d P X ( x ) = Z IR g ( x ) d P X ( x ) . Therefore the desired result holds for indicators. (b) Using the linearity of the expectation, check that this identity holds whenever g ( x ) is a (non negative) simple function on (IR , B ). ANS: Let g ( x ) be a nonnegative simple function. Then there exists constants c 1 , . . . , c n 0 and Borel sets B 1 , . . . , B n such that g ( x ) = n X i =1 c i I B i ( x ) . Hence, by the linearity of the expectation and the integral (which denotes an expectation of g ( x ) on (IR , B , P X )), we have E [ g ( X )] = E " n X i =1 c i I B i ( X ) # = n X i =1 c i E [ I B i ( X )] = n X i =1 c i Z IR I B i ( x ) d P X ( x ) (by part (a)) = Z IR n X i =1 c i I B i ( x ) d P X ( x ) = Z IR g ( x ) d P X ( x ) . (c) Combine the definition of the expectation via the identity (1.2.2) with Monotone Convergence to deduce that (1.4.1) is valid for any nonnegative Borel function g ( x ). ANS: Let g ( x ) be a nonnegative Borel function. Then there exists a sequence { g n } of simple functions such that g 1 0, g n g n +1 , and g n ( x ) g ( x ) as n (for example, take g n ( x ) = f n ( g ( x )) for f n ( ) of Proposition 1.2.6). Hence, E [ g ( X )] = lim n E [ g n ( X )] (Monotone Convergence for g n ( X ( ))) = lim n Z IR g n ( x ) d P X ( x ) (part (b)) = Z IR g ( x ) d P X ( x ) (Monotone Convergence for g n ( x )) 1 (d) Recall that g ( x ) = g + ( x ) g ( x ) for g + ( x ) = max( g ( x ) , 0) and g ( x ) = min( g ( x ) , 0) nonnegative Borel functions. Thus, using Definition 1.2.25 conclude that (1.4.1) holds whenever E  g ( X )  < ....
View
Full
Document
This document was uploaded on 04/12/2010.
 Spring '09

Click to edit the document details