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Unformatted text preview: o facts shows E1 V0 = 1. Reintroducing our
old hitting time T0 = min{n > 0 : Xn = 0} and noting that on our ﬁrst step
we go to 0 or to 1 with probability 1/2 shows that
E0 T0 = (1/2) · 1 + (1/2)E1 V0 = 1
Summarizing the last two paragraphs, we have
III. When p = 1/2, P0 (T0 < 1) = 1 but E0 T0 = 1. Thus when p = 1/2, 0 is recurrent in the sense we will certainly return, but
it is not recurrent in the following sense:
x is said to be positive recurrent if Ex Tx < 1. If a state is recurrent but not positive recurrent, i.e., Px (Tx < 1) = 1 but
Ex Tx = 1, then we say that x is null recurrent.
In our new terminology, our results for reﬂecting random walk say
If p < 1/2, 0 is positive recurrent
If p = 1/2, 0 is null recurrent
If p > 1/2, 0 is transient
In reﬂecting random walk, null recurrence thus represents the borderline
between recurrence and transience. This is what we think in general when we
hear the term. To see the reason we might be interested in positive recurrence
recall that by Theorem 1....
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 Spring '10
 DURRETT
 The Land

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