This preview shows page 1. Sign up to view the full content.
Unformatted text preview: lity. (The denominator is the same, while the numerator is
rearranged.) Since there are n ways to choose the j draws on which we get
j
red,
✓
◆
j
1
P Xn =
=
for 1 j n + 1
n+2
n+1
and it follows that the distribution of the limit X1 is uniform. Example 5.15. Branching Processes. In this system introduced in Example
1.8, Zn is the number of individuals in generation n and each gives rise to
an independent and identically distributed number of individuals with mean
0 < µ < 1 in generation n + 1. If p(x, y ) is the transition probability of the
Markov chain
X
1X
µx
p(x, y )f (y, n + 1) = n+1
p(x, y )y = n+1 = h(x, n)
µ
µ
y
y
so using Theorem 5.5 we see that Wn = Zn /µn is a martingale.
Using this we can rederive some of the facts proved in Example 1.52, and
prove at least one new one.
Subcritical. If µ < 1 then P (Zn > 0) µn EZ0 ! 0 as n ! 1
Proof. Since Zn /µn is a martingale, EZn = µn EZ0 . Using this with P (Zn
1) EZn gives the desired result.
Critical. Let pk be the probability an individual h...
View
Full
Document
This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).
 Spring '10
 DURRETT
 The Land

Click to edit the document details