I
I.
I
I
I
I
I
I
I
NECESSARY CONDITIONS FOR ALMOST SURE EXTINCTION OF A
BRANCHING PROCESS WITH RANDOM ENVIRONMENT
by
Walter L. Smith
University of North Carolina
Institute of Statistics Mimeo Series No. 549
September 1967
I_
I
I
I
I
I
I
I
••
I
This research was supported by the Department of the
Navy, Office of Naval Research. Grant NONR-855(09).
DEPARTMENT OF STATISTICS
University of North Carolina
Chapel Hill, N. C•
- - - : :....... 0/ . . . .
I
I.
I
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
'I·
I
1.
Introduction
In this note we shall be considering what may be called a branching
process with random environment (B.P.R.E.); it will be a sequence {Zn} of
random variables forming a particular sort of Markov chain.
Let {r,; } be a sequence of independent and identically distributed
n
random variables, to be known as environment variables, taking values in
some (possibly abstract) space E.
Suppose that every point r,;EE has associ-
ated with it a probability generating function
~r,;(s),
0
~
s
<
1, of an
integer-valued random variable.
The B.P.R.E. develops as follows:
Zo· 1; Zn+l is the total number of
offspring resulting from Z parents, each such parent having a random number
n
of offspring governed by the p.g.f.
~r,;
(s) independently of other parents
n+l
and of the environment variables other than r,; +.
n 1
This model is clearly the
same as the classical Galton-Watson branching process except that we allow
family-size distributions to vary stochastically from generation to generation.
However, all families of a given generation are governed by the same
distribution of family size.
Thus the separate family trees springing from
different parents in a given generation, which are independent in the
classical Galton-Watson process, a fact which renders the classical GaltonWatson process so tractable, are dependent in the B.P.R.E.
The B.P.R.E. will be discussed more fully elsewhere (Smith and
Wilkinson, 1967).
In the present note we are concerned with proving one
theorem which, taken with Theorem A below (whose proof will
b~
given in
the aforementioned reference), settles at an acceptable level of generality
the question:
become extinct?
under what circumstances will the B.P.R.E. almost surely
I
II
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
.-
I
2
At this point we need some definitions.
We shall suppose the means of
the family size distributions
1 - <l>c;
~.
n
lim
def sll
(s)
n+l
1 - s
n •
1,2,···,
are independent and identically distributed proper random variables
p{~
n
<~}
= 1).
(l.~.
We shall also write
nn • def
<l>c;
n+l
(0)
for the probability that a parent of the nth generation shall have no offspring.
We shall additionally suppose that {n } constitutes a sequence of
n
independent and identically distributed random variables.
To avoid trivia1-
ity we shall assume P{nn • O} < 1, since the alternative possibility obo
vious1y implies P{Zn > O} • 1 for all n.
We can now quote:
Suppose that 1110g ~
Theorem A
n
>
I
<~.
Then
The B.P.R.E. {Zn } will almost surely become extinct,
a)
p{Z
n
O}
-+
If. t log
b)
~
n
> 0
e log
~ < 0;
n -
and if, additionally,
1110g(l - n
(Al)
then p{Z
0 as n -+~, if
n
n
>
b.~.
)1
<~,
O} tends to some strictly positive limit as n
extinction is not almost sure.
-+ ~, b.~.
I
I.
I
I
I
I
I
I
I
3
The theorem we prove in this paper is as follows:
Theorem 1
If there is a c > 0 such that p{Z >
n
extinction is ~ almost certain, and if ellog ~nl < ~, then it is necessary
that the following two conditions both hold:
(i)
t log
~
n
>
0
The method of proof in the present note is
, log
~
<
n -
I
I
It will be noted
0; thus two different methods are available for proving this
tions have, almost surely, finite mean values.
••
different from that
that both theorems prove that extinction of the B.P.R.E. is almost sure if
Ie
I
I
I
I
I
~tirely
used by Smith and Wilkinson (1967) to prove Theorem A.
partial result.
I
O} > c for all n, i.e. if
--
All this work, it should be noted, assumes the family size distribu-
cases when this assumption is invalid.
Work remains to be done for
In this connection we close this
note with a specific example of such a situation, and.show that, for this
special example, the necessary and sufficient conditions for almost certain
extinction are quite different from those obtained in Theorems A and 1.
2.
Proof of Theorem 1.
For every A > 0, let us write
We need two preparatory lemmas.
:1
I.
I
I
I
I
I
I
I
4
Lemma 2.1
If 0 < c <
m
t 0 <
I
I
I
I
'I·
I
m~
<
then
(2.1)
if and only if ~1
Proof
1 and ellog(l - ~)
>
~(A)
The function
~1
<
~
1t
~1 >
Now t if 0
<
n
I
< ~.
is non-increasing and so the series (2.1) must
obviously diverge if
1.
Therefore suppose
[
If we set x
I
mo
~1 dr <
1
= c-1 ~1r
~1 >
1.
1t
-1 r
nC
-1 r
nC .
~1 dr.
log(l/n) we find
-x dx
e
< (log ~1)
x
Ie
I
I
I
~1
E
r-l
m
I
-1 r
C
n
~1 <
c- 1log(1/n)
Let us write G(u)
= P{nu
~
ties and the fact that
u} for the d.f. of nu'
r
~ ~
e-x
e-
X
~
From the above inequali-
10g(1/y)
y
as
y~Ot
it follows on taking expectations that (2.1) converges if and only
if
1
I
I
(2.2)
for some small
E >
O.
10g{10g(1/u)}dG(u) <
But if 1 -
E <
m
u < It we can find 0 > 0 such that
(1 - u) < log(l/u) < (1 + 0)(1 - u).
I
••I
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
'
••
I
5
Thus (2.2) will hold if and only if
1
l
(2.3)
J
log{(l _ u)}dG(u)
< ~t
1-£
and this result establishes the lemma.
Lemma 2.2
If p
= 'log
~n and we choose any 11 1
t e -A~1 ~2' •• ~
for all sufficiently
Proof
>
e P then
n
1 -Alll
> - e
2
lar~.
By the weak law of large numbers.
n-+~
in probability.
Thus if PI > P.
n
P{JI log
~n <
np 1 } ....... 1.
n
-+ ~
and so we can find nO(P 1 ) such that
This implies that. for any A > 0. if we set 11 1
= e P1 •
n
1 e -A1l1
2
•
> -
Thus the lemma is proved.
n > n •
-
0
I
I.
I
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
I·
I
6
We are now ready to prove the theorem.
We begin by observing that
(2.4)
It is an easy exercise to verify that
~
~
1 2
...
~
n
Thus
Plainly P{Z
n
>
O} is a non-increa.ing function of n.
extinction is not almost sure.
p{Z
n
>
O}
>
c for all n.
Let us assume that
Then there must be a c > 0 such that
It then follows that
(2.5)
• • -I; •
n.
Let A > 0; by familiar convexity arguments
by (2.5).
If we now take expectations with respect to
we find
(2.6)
for all large n and any
~l
p
> e •
~1'~2""'~n
and use Lemma 2.2
I
I.
I
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
••
I
7
The proof of (2.6) has depended only on the environment variables
~1'~2'···'~n
n =
n
ep
and variables related thereto.
(0) •
+_
n 1
~
for all n
~
These are all independent of
Thus in (2.6) we may take e -A = n and deduce
n
nO.
But (2.4) implies
n
P{Zn+1
O}
>
= r~ll{l
- ~(Zr) IZ r
>
O},
so we discover that
n
o<
for all n.
c < rUn
o
{1 -
1
2
~(c
-1
n
~1)}'
By a well-known result on infinite products this implies the
convergence of
(2.8)
If p
<
0 we can choose
~1
<
1 and obtain a contradiction with Lemma 2.1.
Thus if p < 0 extinction must be almost sure.
If p
>
0 then (2.8) converges for some
we must have
tI 10g(1-nn ) I
~1
>
1.
Lemma 2.1 then shows that
<~.
Thus the theorem is proved except for the case p = O.
n
Sn
= ~1
log
~r'
and note that
> e-
A P{S
n
< Ole
In this case we set
I
I.
I
I
I
I
I
I
I
8
In place of (2.6) we then have
> O} > e-). P{S
< O},
and in place of (2.7) we obtain
'{~(Z
n
) Iz
>
n
O}
> ~(l) P{S
-
<
n -
The infinite product argument then shows that P{Z
n
O}.
>
O}
> C >
0 for all n
requires the convergence of
co
(2.9)
1: P{S <
n= 1
n -
a}.
But it is known from renewal theory that (2.9) can. only converge if
Clog
~
r
> O.
Thus the proof is completed.
An example with infinite mean family sizes.
Ie
3.
I
I
I
I
I
I
I
variables
I·
I
n
Suppose a is a real constant, 0
{~
n
<
a
<
1, an4 that the environment
} are real and such that, almost surely, 0 <
~
n
< 1 for all n.
Let us then set
(s) .. 1 - ~
It is not difficult to show that
~
n
~n
(1 - s)a, 0 < s < 1.
(s) is indeed a p.g.f. and that the mean
of the associated probability distribution is infinite.
Furthermore, elemen-
tary computation will show that these generating functions have a very
handy property:
(s»
I
I.
I
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
I·
I
9
By repeated use of this algorithm we have that
Thus, if IT (s) is the p.g.f. of Z , we have
n
n
Let us set
IT (s)
n
-+
~(A)
=
A
I~.
Then we see from the last equation that
1 if and only if the infinite product
diverges to zero.
This will happen if and only if
ClO
~
n-l
n
[1 - ~(a )]
=
ClO.
Arguments similar to the ones employed in proving Lemma 2.1 then show that
extinction is almost sure if and only if
or, in other words, if and only if
4' log+ 110g (1 - 4>~ (0) I = ClO.
1
Thus, for almost sure extinction, the probabilities
~~
(0) of no offspring
n
from a given parent have got to be "much nearer" unity, in a probabilistic
sense, than would be the case in the situation covered by Theorem 1.
I
I.
I
'1
I
I
I
I
I
Ie
I
I
I
I
I
I
I
I·
I
10
Reference
W. L. Smith and W. E. Wilkinson, (1967), "On branching processes with
random environments," to appear.