A random sum

1
A random sum

• X1,...,Xn iid GX, N GN
• has pgf
• N is called a stopping time.

2
Simple random walk

• p0(n)P(Sn0)
• f0(n)P(S1?0,...,Sn-1?0,Sn0)P(T0n)
• where T0 is the random time of first return to 0.
• Let
• Theorem
• P0(s)1P0(s)F0(s)
• P0(s)(1-4pqs2)-1/2
• F0(s)1-(1-4pqs2)1/2

3
Proof

• P0(s)1P0(s)F0(s)
• P0(s)(1-4pqs2)-1/2
• 0

4
Return to 0

• P(ever return to 0) F0(1) 1-(1-4pq)1/2
• 1- q-p
• If qp so return is certain, expected time until
  return is
• The random walk is recurrent if return is
  certain, transient otherwise.
• The fair random walk is recurrent in two
  dimensions (the drunkards walk) but not in three
  or more.

5
A branching process

• Survival of family names
• Nuclear pile criticality
• Epidemics

Z01
Z12
Z24
Z33
6
Some facts

• A branching process is a Markov process
• EZn E(X)n
• VarZn
• Let unP(Zn0).
• Then unGX(un-1).
• u0?

7
u1
u2
8
The extinction probability

• which is the smallest non-negative root of
  GX(s)s.
• It is 1 if E(X) 1.