HW3_F09 - Stat 219 - Stochastic Processes Homework Set 3,...

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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 non-negative 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 non-negative Borel function g ( x ). ANS: Let g ( x ) be a non-negative 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) non-negative Borel functions. Thus, using Definition 1.2.25 conclude that (1.4.1) holds whenever E | g ( X ) | < ....
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HW3_F09 - Stat 219 - Stochastic Processes Homework Set 3,...

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