6.262.PS12.sol

# 6.262.PS12.sol - 6.262 Discrete Stochastic Processes Spring...

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Problem Set 12 — Solutions Not due. Problem 1 (Exercise 7.28) a First note that E( Y n | Z n + m - 1 ,Z n + m - 2 ,...,Z 1 ) = E( Z n + m - Z m | Z n + m - 1 ,Z n + m - 2 ,...,Z 1 ) = E( Z n + m | Z n + m - 1 ,Z n + m - 2 ,...,Z 1 ) - E( Z m | Z n + m - 1 ,Z n + m - 2 ,...,Z 1 ) by the linearity of the expectation, since, as Z n is a martingale, both terms are necessarily ﬁnite. (Essentially, Z n being a martingale rules out the fact that E( Z n + m | Z n + m - 1 ,Z n + m - 2 ,...,Z 1 ) = E( Z m | Z n + m - 1 ,Z n + m - 2 ,...,Z 1 ) = , in which case E( Z n + m - Z m | Z n + m - 1 ,Z n + m - 2 ,...,Z 1 ) may exist but the diﬀerence of the two expectations is undeﬁned.) Since E( Z n + m | Z n + m - 1 ,Z n + m - 2 ,...,Z 1 ) = Z n + m - 1 (as Z n is a martingale) and E( Z m | Z n + m - 1 ,Z n + m - 2 ,...,Z 1 ) = Z m (as Z m is included in the condition on Z n + m - 1 ,Z n + m - 2 ,...,Z 1 ), it follows that E( Y n | Z n + m - 1 ,Z n + m - 2 ,...,Z 1 ) = Z n + m - 1 - Z m . b First note that specifying Z n + m - 1 ,...,Z 1 entirely speciﬁes Y n - 1 ,...,Y 1 but not vice-versa (why?). However, we can see that specifying Z n + m - 1 ,...,Z 1 is equivalent to specifying Y n - 1 ,...,Y 1 ,Z m ,Z m - 1 ,...,Z 1 . By Problem 2-a), we therefore have E( Y n | Y n - 1 ,...,Y 1 ) = E(E( Y n | Y n - 1 ,...,Y 1 ,Z m ,Z m - 1 ,...,Z 1 ) | Y n - 1 ,...,Y 1 ) = E(E( Y n | Z n + m - 1 ,Z n + m - 2 ,...,Y m +1 ,Z m ,Z m - 1 ,...,Z 1 ) | Y n - 1 ,...,Y 1 ) = E( Z n + m - 1 - Z m | Y n - 1 ,...,Y 1 ) by part a) = E( Y n - 1 | Y n - 1 ,...,Y 1 ) by deﬁnition = Y n - 1 c E( | Y n | ) = E( | Z n + m - Z m | ) E( | Z n + m | + | Z m | ). Since Z n is a martingale, it follows that E( | Z k | < for all k , and so E( | Y n | ) E( | Z n + m | ) + E( | Z m | ) < . Problem 2 a We show the tower property assuming X,Y and Z have densities. The proof adapts easily to the case where the three random variables are discrete by considering the corresponding probability mass functions. The most general case is discussed afterwards. 1

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## This note was uploaded on 10/21/2010 for the course EE 5581 taught by Professor Moon,j during the Spring '08 term at Minnesota.

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6.262.PS12.sol - 6.262 Discrete Stochastic Processes Spring...

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