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Handout on a Critical Branching Process

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Problem. Consider a branching process with the following probabilities: p_j=P(N₁=j)=(½)^j+1,  j=0,1,2,.... Find the probability e_j that the process dies out by generation j,  and also find the distribution of N_j.
Compute the conditional istribution of N_j conditioned on the event {N_j > 0}, and find the conditional expected number of members of generation j, that is E(N_j | N_j > 0).

Solution
First, note that this is a critical branching process - compute the average number of particles
E N₁=1/2×0 + 1/4 ×1 +1/8 ×2+1/16 × 3+ .... = 1  (exercice!)
Hence the process must eventually die out.

Next, observe that
    G(s)=1/2 + 1/4 s+1/8 s²+1/16 s³+ .... = 1/2 ( 1+ [s/2] +1/8 [s/2]²+1/16 [s/2]³+....=1/2 (1-s/2)^-1=1/(2-s)

Therefore,
    G₂(s)=G(G(s))=(2-s)/(3-2s)
    G₃(s)=G(G(G(s)))=(3-2s)/(4-3s)
and so on - by induction it is easy to show that
G_j(s)=[j-(j-1)s] / [(j+1)-js]
Hence
jG_j(s)-(j-1) = (j+1-js)^-1 = (j+1)^-1 (1 - r_js)^-1 =  [1+(r_js)+(r_js)²+(r_js)³+...] / (j+1)
where
r_j=j/(j+1)
Therefore,
G_j(s)=[j²+ (r_js)+(r_js)²+(r_js)³+... ] / (j+j²)
Now we can "decode" the p.m.f. of N_j using the p.g.f. Namely:
e_j = P(N_j=0)=j² / (j+j²)  = j / (j+1)
which approaches 1.
Moreover,
P(N_j=1)=(r_j)  / (j+j²) 
    P(N_j=2)=(r_j)² / (j+j²) 
and so on, thus giving
P(N_j=k)=(r_j)^k  / (j+j²) 
for k=1,2,3,....

Finally, since
    1 - P(N_j=0) = sum ( P(N_j=k), k=1,2,3,... )  = sum ( (r_j)^k  / (j+j²) , k=1,2,3,...)
                        =1 / (j+j²) × 1 / (1-r_j) = 1 / (j+1)
we have
P(N_j=k | N_j>0 )=(r_j)^k  / (j+j²)  / P(N_j>0) = (r_j)^k  / j
for k=1,2,3,.... and hence
E (N_j=k | N_j>0 ) = sum ( P(N_j=k | N_j>0 ) × k , k=1,2,3,...) = sum ( k (r_j)^k  / j , k=1,2,3,...) = j+1

Thus the conditional (given survival) expected number of particles goes to infinity!

Note: useful formula:
    sum ( a^k × k , k=1,2,3,...)  = a sum ( a^k-1 × k , k=1,2,3,...) = a sum ( a^k , k=1,2,3,...)' =
                                               = a sum ( a^k , k=0,1,2,3,...)' =  a ( (1-a)^-1 )' = a / (1-a)²