assignment9

# Since 0 is arbitrary all sample paths other than a set

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Unformatted text preview: lim t→∞ � �� n P1j = 0 n≥t j =1 This says that for every � > 0, there is a t suﬃciently large that the probability of ever entering states 1 to � on or after step t is less than �. Since � > 0 is arbitrary, all sample paths (other than a set of probability 0) never enter states 1 to � after some ﬁnite time. 1 Since � is arbitrary, limn→∞ Xn exists WP1 and is either 0 or ∞. By deﬁnition, it is 0 with probability F10 (∞). b) Since Xn is the sum of a random number (Xn−1 ) of IID random variables each of mean Y and variance σ 2 , we have 2 V ar(Xn ) = E[Xn−1 ]σ 2 + Y V ar(Xn−1 ) = E[X0 ]Y =Y n−1 2 2 σ + Y V ar(Xn−1 ) n−1 2 2 σ + Y V ar(Xn−1 ) n−1 and X0 = 1. For Y �= 1, we use Where we used the facts that E[Xn−1 ] = E[X0 ]Y induction on n to establish the desired result.The basic step (n = 1) is 2 2 V ar(X1 ) = σ = σ Y Assume that V ar(Xn−1 ) = σ 2 Y equation V ar(Xn ) = Y n−2 (Y n−1 2 n−1 2 − 1)/(Y − 1). Then...
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## This document was uploaded on 03/19/2014 for the course EECS 6.262 at MIT.

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