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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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 Spring '11
 RobertGallager

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