Example 152 branching processes consider a population

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Unformatted text preview: o facts shows E1 V0 = 1. Reintroducing our old hitting time T0 = min{n &gt; 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 &lt; 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 &lt; 1. If a state is recurrent but not positive recurrent, i.e., Px (Tx &lt; 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 &lt; 1/2, 0 is positive recurrent If p = 1/2, 0 is null recurrent If p &gt; 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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