Foundations of Data Science

f(x)
q
p 0
f(f(x)) f(x) x
Figure 4.11: Illustration of convergence of the sequence of iterations f 1 (x), f 2 (x), . . . to q.
m=1 and p 0 < 1, q equals one and for m >1, q is strictly less than one. We shall see
that the value of q is the extinction probability of the branching process and that 1 − q is
the immortality probability. That is, q is the probability that for some j, the number of
children in the j th generation is zero. To see this, note that for m > 1, lim f j (x) = q for
j→∞
0 ≤ x < 1. Figure 4.11 illustrates the proof which is given in Lemma 4.10. Similarly note
that when m < 1 or m = 1 with p 0 < 1, f j (x) approaches one as j approaches infinity.
Lemma 4.10 Assume m > 1. Let q be the unique root of f(x)=x in [0,1). In the limit as
j goes to infinity, f j (x) = q for x in [0, 1).
Proof: If 0 ≤ x ≤ q, then x < f(x) ≤ f(q) and iterating this inequality
x < f 1 (x) < f 2 (x) < · · · < f j (x) < f (q) = q.
Clearly, the sequence converges and it must converge to a fixed point where f (x) = x.
Similarly, if q ≤ x < 1, then f(q) ≤ f(x) < x and iterating this inequality
x > f 1 (x) > f 2 (x) > · · · > f j (x) > f (q) = q.
In the limit as j goes to infinity f j (x) = q for all x, 0 ≤ x < 1.
∑
Recall that f j (x) is the generating function ∞ Prob (z j = i) x i . The fact that in the
limit the generating function equals the constant q, and is not a function of x, says
that Prob (z j = 0) = q and Prob (z j = i) = 0 for all finite nonzero values of i. The
remaining probability is the probability of a nonfinite component. Thus, when m >1, q
is the extinction probability and 1-q is the probability that z j grows without bound, i.e.,
immortality.
99
i=0