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Unformatted text preview: c when i 0 p) and setting ⇡ (0) = c, we have
✓ p
1 p ◆i (1.29) There are now three cases to consider:
P
p < 1/2: p/(1 p) < 1. ⇡ (i) decreases exponentially fast, so i ⇡ (i) < 1, and
we can pick c to make ⇡ a stationary distribution. To ﬁnd the value of c to
make ⇡ a probability distribution we recall
1
X ✓i = 1/(1 ✓) when ✓ < 1. i=0 Taking ✓ = p/(1 p) and hence 1 ✓ = (1 2p)/(1
of the ⇡ (i) deﬁned in (⇤) is c(1 p)/(1 2p), so
⇡ (i) = 1 2p
·
1p ✓ p
1 p ◆i = (1 p), we see that the sum ✓)✓i (1.30) To conﬁrm that we have succeeded in making the ⇡ (i) add up to 1, note that
if we are ﬂipping a coin with a probability ✓ of Heads, then the probability of
getting i Heads before we get our ﬁrst Tails is given by ⇡ (i).
The reﬂecting random walk is clearly irreducible. To check that it is aperiodic note that p(0, 0) > 0 implies 0 has period 1, and then Lemma 1.18 implies
that all states have period 1. Using the convergence theorem, Theorem 1.19,
now we see that
I. When p < 1/2, P (Xn = j ) ! ⇡ (j ), the stationary distribution in (1.30).
Using Theorem 1.22 now, E0 T 0 = 1
1
1p
=
=
⇡ (0)
1...
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 Spring '10
 DURRETT
 The Land

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