The work in this report was partially supported by the Office of
Naval Research under Contract No. N00014-67-A-0321-0002.
MULTITYPE BRANCHING PROCESSES IN RANDOM ENVIRONMENTS
by
EDWARD W. WEISSNER
Department of Statistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 664
FEBRUARY 1970
ii
TABLE OF CONTENTS
Chapter
Page
ACKNOWLEDGMENTS
iii
·...
ABSTRACT
.
NOTATION AND TERMS
I
vi
INTRODUCTION AND PRELIMINARY RESULTS
1.
Introduction.
• • . • • •
2.
Definition of the process
3.
Generating functions and
moments of Z(n) •
11
4.
Positivity properties
14
5.
Transience of nonzero states
16
1
4
,.."
II
EXTINCTION PROBABILITIES FOR POPULATIONS
CONSISTING INITIALLY OF ONE PARTICLE
6.
7•
8.
9•
10.
III
iv
· · ·
The dual process . . . . . .
······
Products of random matrices
· ···
A condition for almost certain
extinction . . . . . .
· · ·
·
Extinction probabilities
A condition for nonextinction
··
·
21
24
25
30
43
EXTINCTION PROBABILITIES FOR POPULATIONS
WITH ANY GIVEN INITIAL POPULATION
11.
The ergodic distribution •
REFERENCES
. . ... .. ...... ...
56
62
iii
ACKNOWLEDGMENTS
I would like to thank the many fine teachers who have
helped me throughout my mathematical education.
In
particular, I want to thank Mr. George Lenchner who, ever so
gently, introduced me to the beauty and the magnetism of
mathematics.
He firmly believes in the philosophy, so aptly
stated by Plutarch, that "the mind is not a vessel to be
filled, but a fire to be kindled" and for this, I will be
ever grateful.
Also, I wish to thank Dr. M. H. DeGroot for
introducing me to statistics, Dr. P. Ney for providing
insight into the seemingly dull results of advanced
probability, and Dr. M. R. Leadbetter for guiding me in my
earliest research.
Most of all, however, I want to thank
Dr. Walter L. Smith not only for suggesting this thesis
topic and for his genuine interest in my progress, but also
for sharing with me his enthusiasm and love for mathematics.
To each of these men, I will always be indebted.
I also want to acknowledge the very generous financial
assistance which I have received.
I am indebted to the
Whitehall Foundation and Carnegie Institute of Technology
for undergraduate financial support.
Funds for my graduate
education were provided by a teaching assistantship at the
University of Wisconsin, by a NDEA Title IV Fellowship, by
an assistantship from the Department of Statistics at the
University of North Carolina, and by a summer grant from the
Office of Naval Research.
iv
ABSTRACT
Consider a population of particles and k fixed particle
classifications labeled type 1, type 2,
... , type k.
Suppose that to each classification there corresponds
exactly one k-variate p.g.f. and that each particle of the
population is classified by one and only one type.
Further,
assume that after each unit of time, each particle,
independently of the other particles, splits or disintegrates into particles of several types, in accordance with
the k-variate p.g.f. which corresponds to the parent
particle's type.
Given these hypotheses, the probability
of the population ever becoming extinct is known.
This
process, in fact, is the well-known multi type Galton-Watson
process.
In this work, we remove the restrictive assumption
that particles of the same type always divide in accordance
with the same k-variate p.g.f.
Instead, we assume that at
each unit of time, Nature chooses a k-vector of k-variate
p.g.f.s from a class of k-vectors of k-variate p.g.f.s,
independently of the population, past and present, and the
previously selected k-vectors, which is then assigned to the
present population.
Each particle of the present population
then splits or disintegrates, independently of the others,
in accordance with the k-variate p.g.f. assigned to its
v
classification.
We call this process a multi type branching
process in a random environment
(~ffiPRE).
When there is only one particle classification,
necessary and sufficient conditions for almost certain
extinction of the MBPRE are known [Smith and Wilkinson
(Ann. Math. Statist.
Statist.
~~
~
2136-2140)].
814-827) and Smith (Ann. Math.
In this work, we give some
necessary and some sufficient conditions for almost certain
extinction of the MBPRE when there are at least two
particle classifications.
To obtain these results, we use
Jensen's inequality, the dual process suggested by Smith
and Wilkinson (Ann. Math. Statist.
40
results on products of random matrices.
814-827), and some
Since our theorems
require the user to evaluate limits of sequences of
products of random matrices, we include many corollaries
which involve simpler, though less general, conditions.
Further, when Nature has only a finite number of choices
and when the class of moment matrices of the k-vectors of
k-variate p.g.f.s enjoys a special property, we give
conditions which are nearly necessary and sufficient for
almost certain extinction of the MBPRE.
vi
NOTATION AND TERMS
Notation
the nth reference listed in the References
[n]
the abbreviation for branching process in a
BPRE
random environment (p. 2)
~PRE
the abbreviation for multi type branching process
in a random environment (Definition 2.1, p. 7)
the k-vector (1, 1, 1, ... , 1)
1
~
the k-vector whose jth coordinate is one and whose
other coordinates are zero
j=a(b)a+Nb
notation for j = a, a+b, a+2b,
a,b
~
••• , a+Nb where
0 and N is a positive integer
j=a(b)-
notation for j = a, a+b, a+2b,
••• where a,b > 0
T
the set of all k-vectors whose coordinates are
?
nonnegative integers
an environmental random variable
~n
(Assumption Al, p. 5)
e
the environmental space (Assumption Al, p. 5)
m.p.g.f.
the abbreviation for multivariate probability
generating function(s)
¢(8,s)
~
~
the k-vector of m.p.g.f. associated with the
environmental state 8 (Assumption A2, p. 5)
Z(n)
~
the size of the nth generation of the
(Definition 2.1, p. 7)
~PRE
vii
(P(~,£))
the one step transition matrix of the MBPRE
(Defintion 2.1, p. 7)
~(e)
,...
the matrix of moments of the k-vector of m.p.g.f.
~(e,s)
associated with the environmental state e
'"
,...
(p. 10)
the nonrandom matrix of moments of the MBPREi
(p. 11)
Z(O)
the p.g.f. of Z(n) given that ,...
'"
=
a
,...
(Definition 3.1, p. 11)
u > v,...
,...
each component of the vector ,...
u-v
,... is positive
u,... > v,...
each component of the vector u-v is nonnegative
u,...
=
v,...
1£1
each component of the vector u-v is zero
the sum of the absolute values of the coordinates
of the vector u,...
the sum of the absolute values of the entries of
the matrix A
~.
II~II
max.
A (A)
'"
the largest positive eigenvalue of the primitive
1
matrix
q(~)
\A{i,j) \, if A is a square matrix
,...
J
~
(Theorem 4.1, p. 15)
the extinction probability of the MBPRE given that
Z(O)
'"
=
a
,...
(Definition 11.1, p. 56)
~
the k-vector (q{~l)' q(!2)'
~(n)
a random vector of the dual process
.•• , q{!k))
(Definition 7.1, p. 24)
log+ x
max (O, log x), if x is a positive number
M(C,N)
a special class of matrices
(Definition 8.1, p. 27)
viii
r
a critical parameter associated with products of
random matrices (Sections 8, 9, and 10)
the X(n)
process converges in distribution to this
,..,
random k-vector (Remark 11.1, p. 57)
F(~)
the ergodic distribution function
(Definition 11.2, p. 57)
Term
Definition
Page
7.1
24
ergodic distribution
11. 2
57
extinction probability
11.1
56
impartial MBPRE
4.3
16
insecure MBPRE
5.2
18
primitive matrix
4.2
15
singular MBPRE
9.1
42
transient state
5.1
16
dual process
CHAPTER I
INTRODUCTION AND PRELIMINARY RESULTS
1.
Introduction
Consider a population of particles and k fixed particle
classifications labeled type 1, type 2,
•.. , type k.
Sup-
pose that to each classification there corresponds exactly
one k-variate probability generating function and that each
particle of the population is classified by one and only one
type.
Further, assume that after each unit of time, each
particle, independently of the other particles, splits or
disintegrates into particles of several types, in accordance
with the k-variate probability generating function which
corresponds to the parent particle's type.
Given these
hypotheses, the probability of the population ever becoming
extinct is known.
This process, in fact, is the well-known
multi type Galton-Watson process described by Harris
[8, p. 34].
If k=l, the process described is the classical
Galton-Watson process in which each particle is of the same
type and each particle splits or disintegrates in accordance
with the same given probability generating function.
Recently, this model has been generalized by
2
Wilkinson ([11],
[12]) and Smith ([10],
[11]).
Inhis
thesis, Wilkinson removed the restrictive assumption that
the particles always diviae in accordance with the same
probability generating function.
Instead, he proposed that
at each unit of time, Nature be allowed to choose a
probability generating function from a class of probability
generating functions, independently of the population, past
and present, and the previously selected probability
generating functions, which would then be assigned to the
present population.
Each particle of the present population
would then split, independently of the others, in accordance
with this probability generating function.
This process,
called a branching process in a random environment (BPRE),
is, of course, much more applicable than the Galton-Watson
process.
Moreover, Wilkinson's search for necessary and
sufficient conditions for almost certain extinction seems
to be not only elegant, but also fairly complete.
Further,
the conditions are almost as simple to verify as those for
the Galton-Watson process.
The search for necessary and
sufficient conditions for almost certain extinction of the
BPRE has been completed now by Smith [10].
It is the purpose of this work to consider the
mUltitype branching process in a random environment (MBPRE)r
that is, the BPRE process for the case k>2.
As Wilkinson
did with the BPRE process, we endeavored to find necessary
and sufficient conditions for almost sure extinction of the
population.
However, it appears that the determination of
3
the probability of extinction of the MBPRE is not a simple
extension of the techniques used by Wilkinson.
Where he
was able to consider products of nonnegative random
variables, the rlliPRE yielded products of nonnegative random
matrices.
Where he was able to evaluate limits of sequences
of products of random variables by applying the strong law
of large numbers, we could only find existence theorems
for the limits of sequences of products of random matrices
which were of little immediate practical value.
Where
Wilkinson used linear approximations to the probability
generating functions of the BPRE to excellent advantage, we
were forced to use higher order approximations to the
k-variate probability generating functions of the 11BPRE
which in some cases gave excellent results, but in other
cases gave no results.
Many of these difficulties are, in
fact, stimulating research problems in themselves.
Thus,
we shall only be able to give some conditions for almost
sure extinction and nonextinction of the MBPRE.
Let us begin by defining the process mathematically.
We shall then discuss the probability generating functions
of the process, the moments of the process, and the
transience of its states.
Then in Chapter II, we shall
again define the dual process (Smith and Wilkinson [11])
and exhibit some conditions for extinction and nonextinction
of the process.
By including many corollaries and some
results of Furstenberg and Kesten [5], we will endeavor to
make the rather complex conditions more applicable.
4
Finally, in Chapter III, we shall introduce the ergodic
distribution which is associated with the process and which
gives us virtually all the information we require about
extinction of the MBPRE.
As we proceed, we shall observe,
through the many remarks, many of the strengths and weaknesses of the various approaches and methods.
2.
Definition of the process
Notation.
We shall denote vectors and matrices by
lightface letters underscored with a wavy line.
Unless
specifically noted, we will assume that all vectors
(matrices) are of order k (kxk), where k is the fixed
positive integer, greater than one, which corresponds to the
number of distinct types.
A vector premultipling
(postmultiplying) a matrix will be considered a row (column)
vector; otherwise we do not distinguish between row and
column vectors.
r
"'"
=
Further, if r is any real number, let
(r,r, •.. ,r) and ifj is any integer, l<j<k, let e. denote
-
-
""'J
the vector whose jth component is 1 and whose other
components are zero.
Also, we shall denote the jth
component of the vector ,...,a(n) by a (j , n)
for j
=
1,2, ••. ,k and
'th column of the matrix
the entry in the ith row and J_
A(n) by A(i,j,n) for i,j
"'"
=
1,2, .. .,k.
We shall denote the underlying probability space by
(Q,A,P), where Q is some abstract set, A is a sigma-field
of subsets of Q, and P is a probability measure defined on
5
A.
If m is any positive integer, let B
m denote the Borel
m
sets of the real Euclidean space R .
defined on a measurable space (S,F)
Further, if f is
and is a mapping into
the m-dimensional real Euclidean space (Rm,B ), then f is
m
F-measurable if the inverse image of every set of B
m
is a
member of F.
... ,
For simplicity, let j = a, a+b, a+2b,
a+Nb, for
some positive integers band N, be denoted by j = a(b)a+Nb;
if N = 00, denote by j = a(b)-.
Finall~
let T denote the
set of all k-dimensional vectors whose components are
nonnegative integers.
Assumption AI.
Let {~ , n=O(l)-} be an infinite
--
n
sequence of independent and identically distributed
"environmental" random variables defined on
(~,A,P)
and
taking values in e, a Borel set of the Euclidean space
m( e)
R
for some positive integer m(e) •
Assumption A2.
Suppose that associated with each
point eEe there is "a vector of mUltivariate probability
generating functions
(m.p.g. f.),
<I>(e,s) = [<I>(l,e,s), <I>(2,e,s), ••. , <I>(k,e,s)],
~
~
~
~
~
k
where s = (s(l), s(2), ... , s(k)), ,....sE[O,l], and
~..;..;,..-,....
(2.1)
= '£.o,e.ET
("
e )
<I>(j,e,s)
P ('J,,~
,....
for j=l(l)k.
n"i=l
k
s ( 1,)a(i)
,
6
Assumption A3.
Suppose that for each fixed ,...,sE[O,l]k,
2(e,~) is a mapping from e into [O,l]k which is Borel
measurable.
Remark 2.1.
If e has at most countably many points,
then Assumption A3 is trivially satisfied.
Remark 2.2.
We may immediately conclude from
Assumption A3 that for fixed ,t(l),
t(2), ••. , ,...,
t(k)
...,,...,
[¢(j,e, £(i», i,j = l(l)k], a mapping from
is Borel measurable.
t(k)
,...,
e
k
[0,1]
2
into [0,1] k ,
E
.. . ,
Thus, for fixed ,...,
tel), ,...,
t(2),
E [O,l]k, Assumption Al implies that
{[¢(j,l;;n' !(i»,
i,j = l(l)k],
n=O(l)-}
is an infinite sequence of independent and identically
distributed random arrays defined on (n,A,p).
More specifi-
cally, for fixed sE[O,l]k, {¢(l;; n ,s), n=O(l)-} is an infinite
,...,
"*"I
#"'OJ
sequence of independent and identically distributed random
vectors defined on (n,A,p).
Remark 2.3.
2' (e l'
.2 (e 2'
For fixed ,...,sE[O,l]k
.•• , 2 (en' Z)
••. ) is a mapping from e
n
into [O,l]k which is Borel measurable for n=l(l)-.
demonstrate this, let j be any integer such that
To
l~j~k
and define f(e,~) on ex[O,l]k by f(e,~) = ¢(j,e,~).
Clearly f(e,t)
,...,
is a mapping into [0,1].
Assumption A2 implies that f(e,t)
,...,
For fixed eEe,
is a mapping from
[O,l]k into [0,1] which is continuous.
For fixed ,...,
tE[O,l]k,
7
Assumption A3 is that f(8,t)
,..,
is a mapping from e into
[O,ll which is Borel measurable.
f
8
Thus, since each section
is continuous and each section f t is Borel measurable,
k
f (8 ,~) is Borel measurable on ex [0,1 1 •
is also Borel measurable.
the vector (8 1 ,
!(82'~»
Hence,
1 (8, i)
Now fix ,..,
s€[O,llk and consider
2
defined on e and taking values
in 0x[0,llk.
Clearly this vector is Borel measurable
and therefore
2(81'2(82'~»
is Borel measurable.
A simple
induction argument now completes the proof.
Remark 2.4.
For fixed ,..,s€[O,llk,
2(1;0' 2(1;1' •.• , 2(1;n'~) •• ) and !(1;n' 2(1;n-l' .. , !(1;O'~)")
are random vectors defined on (n,A,p) for n=O(l)-.
In the next few sections the importance of these
remarks will be self-evident.
.Without these results, the
measurability of many random variables would be suspect.
Definition 2.1.
The mUltitype branching process in a
random environment (MBPRE) is defined to be a temporally
homogeneous vector Markov chain {Z(n),
n=O(l)-} whose states
,..,
are vectors in T.
We shall always assume that Z(O) is
""
nonrandom and we shall interpret the ith component of ~(n),
Z(i,n), as the number of type i particles in the nth
generation.
The transition law for the process is as follows.
~,£ € T, ~€[O,llk and n=O(l)-, define the matrix (P(~,~»
by
For
8
P(~,£)
(2.2)
= coefficient of
TIi~l
s(i)b(i) in
E{TIj~l cj>(j,c;n,~)a(j)}.
Remark 2.2 implies that
of n.
Plainly
P(~,£)
P(~,£)
> 0 for
is independent of the choice
~,B
E T and since
(2.3)
it follows that Lb E T P(a,b) = 1 for each aET. We can now
.....
define the vector Markov chain {~(n), n=O(l)-} by choosing
I">J
~
""oJ
initial probabilities
Pr{Z(O)
.....
=
~}
=
r
1,
if
Lo,
if
a=c,
..... .....
for some £ET,
and defining
.e (i),
Pr { ~ ( i) =
where .....
a(i)
i = 0 ( 1) n} = P r {Z ( 0 ) = ~ ( 0) }TI i ~ 1P (~ ( i -1), ~ ( i) )
E T, i=O(l)n.
If Pr{Z(n)
= .....
a} is positive, then
.....
P(a,b) is the transition probability Pr{Z(n+l) = blz(n) = a}.
~
""'J
,.,.""
1"0.1
,.,.""
t"W
While all our results can be derived using
Definition 2.1 it may be extremely helpful if we pause here
to give a more concrete interpretation to the MBPRE.
Consider a population of particles.
Suppose that, at
any time, there are at most k distinct types of particles,
labeled 1,2, .•• ,k, that may be represented in the population.
Suppose, also that after one unit of time each
particle of the population either disintegrates or splits
9
randomly, generating particles of several types.
The
initial population of particles, called the zeroth
generation, _generates particles which comprise the first
generation, the first generation generates particles which
comprise the second generation, and so on.
Since we are
primarily interested in the size of the nth generation
(n=O(l)-), let ,...Z(n) = (Z(l,n), Z(2,n), ... , Z(k,n» denote
the respective number of particles of each type in the
nth generation; that is, Z(j,n) is the number of type j
particles in the nth generation.
By assumption, ,...Z(O)
= ,...a
for some fixed a£T.
,...
Now assume that Nature chooses an environmental state,
~n =~,
independently of the past environmental states and
the population, past and present, which uniquely determines
the stochastic"'manner in which. the particles of the nth
generation will. generate particles of the (n+l)st
generation.
One may think of
~
as being a particular
combination of environmental factors such as heat, pressure,
or light which will affect the splitting or disintegration
of the particles.
Mathematically, this effect is completely
defined by some vector of m.p.g.f., ,...,...
¢(~,s).
If ,...
¢(~,s) is
defined as in Assumption A2, we may interpret the
environmental effect
b£T, then
,...
p(j,~,b)
,...
~
as follows:
(1) if
is the probability that one type j
particle will generate b(i) type i particles, i=l(l)k,
and (2) each particle of the nth generation will split
independently of the other particles present, but in
10
- -
accordance wi th <!> ( z; , s) .
Thus, we assume that the environment passes through
a sequence of states governed by the infinite sequence
{z; n , n=O{l)-} of independent and identically distributed
Then, given z; =Z;, each particle of the nth
random variables.
n
generation splits or disintegrates independently of the
other particles of the nth generation, but in accordance
- -
with the specified vector of m.p.g.f. <!>{Z;,s).
Remark 2.5.
Observe that if 8 contains only one
point, then the MBPRE is precisely the standard multi type
Galton-Watson process.
Let us now consider certain moments associated with
the MBPRE.
Since for arbitrary 8£8 and i any integer
[1 -
{2. 4}
A..
'I'
{i , 8, 1-teJ. } ] / t ,
j=l{l}k,
increases to a nonnegative {possibly infinite} limit, say
~{i,j,8},
which is the mean number of type j particles
produced by one type i particle in the environmental state 8.
Notation.
{~{i,j,8}}.
For 8£8, let
~(8)
be the matrix defined by
Clearly ,..~{8} is of order kxk.
Assumption A4.
finite; that is
Remark 2.6.
Suppose that for each 8£8,
~(i,j,8)
~(8)
is
is finite for i,j = l{l)k.
We can now prove that {~(Z; ), n=O{l)-}
-
n
11
is an infinite sequence of independent and identically
distributed nonnegative random matrices defined on (n,A,p).
First, observe that Assumption A4 and Assumption A3
applied to (2.4) implies that ,..,
~(e), a mapping from
e
into
the nonnegative kxk matrices, is Borel measurable.
Assumption Al then implies that
{~(~
,..,
n ), n=O(l)-} is an
infinite sequence of independent random matrices on
(n,A,p).
Clearly the terms of this sequence are also
identically distributed.
Notation.
~
,..,
~
Let
denote the square matrix defined by
= E{~(~
)} = (E{~(i,j,~ n )}).
,.., n
Plainly, ~ is independent
of n and well-defined, since the entries of ,..,~(~ n ) are
nonnegative for n=O(l)-.
are nonnegative.
3.
Observe that the entries of
~
Some may even be infinite.
Generating functions and moments of
Definition 3.1.
~(n)
For n=O(l)- and ,..,
a€T, let G(n,sla)
,.., ,.., be
the multivariate probability generating function of the
random vector ,..,
Z(n), given that ,Z(O)
. . , ,=. .a.
,
Theorem 3.1.
For n=l(l)-, ,..,
s€[O,l]k, --,..,
and a€T
(3.1)
In accordance with Definition 2.1, G(O,s!a)
,. .", ,. . , =
Proof.
rr'~l
J- s(j)a(j).
Let n be any integer greater than one, choose
a€T, and fix ,..,
s€[O,l] k •
,..,
From equation (2.3), we see that
12
=
bIZ(O)
,."
,."
By the dominated convergence theorem, we obtain (3.1) for
n=2(l)-,
SE
,."
[O,l]k and aET.
,."
To show (3.1) for n = 0,1, we
appeal directly to Definition 2.1.
Remark 3.1.
By repeated application of Theorem 3.1,
we obtain the representation
(3.2)
for n=l(l)-, a£T and s£[O,l]k.
,."
Remark 3.2.
~
n-l
,."
Since the random variables
~O'
~l'
••. ,
are independent and identically distributed, we may
renumber them without affecting the value of the right
hand side of (3.2). Thus, we also have the representation
for n=l(l)-, a£T and sE[O,l]k.
,."
Remark 3.3.
,."
Let N be any fixed positive integer.
Then
{Z(nN), n=O(l)-} is also a temporally homogeneous vector
,."
Markov chain whose states are vectors in T.
Using (3.2), we
observe that if ,.,,.,,
a,b E T and Pr{Z(nN) = a} is positive, then
,."
13
Pr{~{nN+N) = EI~{nN) = ~} = coefficient of rri~l s{i)b{i) in
E{rrj~l
~)
¢(j,so' ;e{sl' ;e{s2' ... , ;e{sN-l'
... )a{j)}
for n=O{l)-.
Now Assumption Al implies that
{(snN' ••. , snN+N-l)'
n=O{l)-} is an infinite sequence of
independent and identically distributed "environmental"
random vectors defined on (Q,A,P) and taking values in eN,
a Borel set of the Euclidean space RNm{e).
implies that for each point eel' e 2 ,
Assumption A2
N
•.. , eN) £ e ,
(3. 4)
is a vector of m.p.g.f. defined on [O,l]k.
-
implies that for each fixed s£[O,l]k,
Also, Remark 2.3
(3.4) is a mapping
from eN into [O,l]k which is Borel measurable.
Thus, a
close review of Definition 2.1 reveals that we may also
consider the temporally homogeneous vector Markov chain
-
{Z{nN), n=O{l)-} as a MBPRE.
E{~ (n+m)
Theorem 3.2.
I ~(n) } =
~(n)
with probability one, for n,m = O{l)-.
Proof.
-
of
~m,
Since the theorem is clearly true if m=O, we
-
begin by setting m=l.
a£T.
-
-
Suppose also that Zen) = a for some
Then, by Definition 2.1, we observe that the m.p.g.f.
~(n+l),
given
~(n)
=
~,
is given by
14
Since
nj~l ¢(j,e,~)a(j) is a m.p.g.f. for each 6£8, it
follows, using the monotone convergence theorem, that we
may evaluate the left hand partials of G(l,s!a)
,.., ,.., with
respect to sCi) at ,..,s=l,
i=l(l)k; in particular,
,..,
lim t + O+ [1 - G(l,
l-t~il~)]/t = E{Ej~l a(j) ~(j,i,~O)}
k
= Ej=l a(j)
~(j,i),
i=l(l)k.
Therefore, E{Z(n+l) Iz(n) = a} = a~ and thus, with
~
~
~
~~
probability one,
(3.5)
E{Z(n+l) IZ(n)} = Zen)
,....,
,.",.,
~,
n=O(l)-.
4"'tJ,.",.,
Now using (3.5), we have that, with probability one,
E{E{Z
(n+m) I Z (n+m-l),
,....,,....,
•.• , Z (n) } I Z (n) }
,.",.,,....,
= E{E{Z
Z (n+m-l) } I ,..,
Z (n) }
,.., (n+m) I ,..,
= E{Z(n+m-l). slz(n)}
'"
=
'" '"
..• =
for n=O(l)- and m=2(1)-.
This completes the proof.
Corollary 3.2.1.
E { Z (n)
'"
IZ ( 0 ) =
,..,
e.}
'" J
=
e. ~n
"'J""
for j=l(l}k and n=O(l}-.
4.
positivity properties
Before considering the transience of the nonzero states,
let us pause to define and state some important properties
of matrices.
Since these properties will be used frequently,
the reader is urged to pay particular attention to the
definitions and conventions.
15
Definition 4.1.
We shall call a vector or a matrix
positive, nonnegative, or
these properties.
then
~
> ~ (E ~~)
Notation.
~
if all its components have
If u and v are vectors or matrices,
"'"
means
"'"
~-~
is positive (nonnegative).
Let absolute value signs enclosing a vector
or a matrix denote the sum of the absolute values of the
elements of the vector or the matrix, respectively.
For
a square matrix A, with real entries, denote the norm of
"'"
~ by
I I~ I I
= maxi Lj
IA (i , j) I .
Definition 4.2.
A nonnegative square matrix
h is
said to be primitive if there is a positive integer N
such that AN is positive.
Theorem 4.1.
~
Let
which is primitive.
Then
be a nonnegative matrix of order m
~
has a positive characteristic
root A that is simple and greater in absolute value than
any other characteristic rooti A corresponds to positive
left and right eigenvectors
nonnegative eigenvectors.
~
and
~
which are the only
Moreover, we have
~n = An~l + ~~ ,
n=l(l)-,
where bl = (u(i)v(j», with the normalization
m
Li=l
u(i)v(i) = 1.
(a) ~1~2 = ~2~1 = Q.i
Hence ~l~l = ~l'
(b)
Furthermore,
It-~I = O(on) for some 0, O<O<Ai
·
"
r A ( i,j) ~ Ai
(c ) f or eac h r, 1~r<m, we h ave m~nl~i~r
£.oj=l
j
(d) if j is a positive integer, then A corresponds to
A
"'"
j
just as A does to ~.
16
For a proof of this theorem the reader
is referred to
Gantmacher [6, p. 65] and Karlin [9, p. 475].
Notation.
If A
,... is a nonnegative matrix of order m
which is primitive, let A(A)
,... be the largest positive
characteristic root of ,...
A.
Definition 4.3.
The MBPRE is said to be impartial
if the nonnegative square matrix M is primitive, where M
,...
is defined by
l-1(i,j)
=
rD'
=
if s(i,j)
0,
1, if s(i,j) is positive or infinite,
and ,...
s is the matrix defined by ,...s
Remark 4.1.
=
E{£
U;o) }.
If the square matrix ,...s is finite, then
the MBPRE is impartial if and only if there is a positive
integer N such that ,...
sN is positive.
Observe that if the MBPRE is impartial, there is a
positive probability that there will be at least one type j
particle,
l~j~k,
generated in the next N generations.
Hence, we expect to find type j particles,
l~j~k,
in the
population frequently; we do not expect the population to
consist only of type iI' type i 2 ,
••• , type i h particles,
h<k, from some generation on.
5.
Transience of nonzero states
Definition 5.1.
The state ,...
aET is said to be transient
17
if pr{!(n) = ~ for some n=l(l)-I!(O) = ~} < 1, or
equivalently if pr{~(n) = ~ infinitely often I!(O) = a} < l.
I'W
Remark 5.1.
P(Q,Q) = 1.
From Definition 2.1 we observe that
~(n)
Thus, with probability one, if
=
Q,
then
Z(m) = Q for m=n(l)-; that is,Q is an absorbing state.
Now if
2
£
T-{Q} and pr{](n) = QI~(o} = ~} is positive,
then by Definition 5.1 and Remark 5.1, we infer that a is
I'W
transient.
Further, since pr{!(n) = £I~(O) = ~} is
nondecreasing as n increases, if
~ £
T-{Q}and
limn+oo Pr{](n) ~ £I~(O) = ~} is positive, then, again, we
can infer that ....
a is transient •
Since we are interested, however, in evaluating
(5.1)
for
~ £
limn+oo Pr{Z(n) = OIZ(O) = a}
~
~
~
~
T-{Q}, it is doubtful that we will always know in
advance whether or not (5.1) is positive.
In fact, the
proof of the well-known result for the almost certain
extinction of the multi type Galton-Watson process presupposes that the nonzero states of T are transient
(Harris [8, p. 38]).
Our approach, however, to evaluating
(5.1) will not require us to study the transience of the
nonzero states first.
But, to illustrate the methods of
the standard type of proof, we shall give a crude condition
under which transience of the nonzero states of T occurs.
Later in Section 6 we shall use this condition to prove a
generalization of the well-known theorem for the almost
18
sure extinction of the multi type Galton-Watson process.
The MBPRE is said to be insecure
Definition 5.2.
if pr{~(n) = QI~(O) =
l}
=
G(n,Qll)
> 0 for some positive
integer n.
Suppose the MBPRE is insecure.
Theorem 5.1.
a
f'IotI
£
If
T-{O}, then a is a transient state.
""-I_1"ittI
Proof.
Choose
Pr{~(n) = QI~(O)
~ £
T-{Q} and suppose that
= ~} = 0 for n=l(l)-.
Then G(n,Ola)
,..., ,..., = 0
for n=l(l)- and we have, using Remark 3.2, that
k
E { IIj=l <j>(j,1;;n-l' 1(1;;n-2' .•• , t(1;;o'~)···)
for n=l(l)-.
a(j)
} = 0
Thus, with probability one,
for n=l(l)- and therefore, with probability one,
for n=l(l)-.
n=l(l)-.
But this violates the hypothesis that the MBPRE
is insecure.
Pr{~(n)
It now follows that G(n,Oll)
,..., ,..., = 0 for
Thus there is a positive integer n such that
= QI~(O)
= ~} is positive.
From our previous
comments, we now conclude that ,...,
a is transient.
Theorem 5.2.
Suppose the MBPRE is insecure.
N is an arbitrary positive integer, then
If
19
(5.2)
Pr{O < l.e(n)
I
< N}
-+-
0,
as n
-+-
00,
and
(5.3)
Pr{O < I!(n)
Proof.
I
< N infinitely often} =
o.
This result follows from Theorem 5.1 and
Feller [4, Vol. I, Chapter 15].
Theorem 5.3.
each
----
Suppose the MBPRE is insecure.
a € T-{O}, there exists a number u(a)
~
~
~
such that for every ~€[O,l)k, G(n,~I~)
as n
-+-
For
€ [0,1]
u(2)'
-+-
00
Choose s€[O,l)k and
Proof.
T-{ O}.
~
~
If N is a
positive integer, then
= G(
I
n,Q~)
I}
U j=lS(J)
k
. b
+ 1.0<lbl<N Pr {
Z,(n)=£
Z(O)=.e
<;'
(j)
~
.
k
b ( .)
+ Llbl>N pr{!(n)=£I~(O)=~}IIj=ls(j) J .
~
Thus,
G(n,Ola) < G(n,sla) < G(n,Ola) + Pr{O < IZ(n)1 < N}
""'J
"..",
"..",
"..",
"..",
"...,
,....,
+ {maxl<i<k sCi) }N.
Since G(n,Ola) is a bounded sequence which is nondecreasing,
~
~
as n increases, there exists a number, say u(a), such that
~
u(a) = limn+oo G(n,Ola).
~
Clearly u(a)
,...
~
~
€ [0,1] and applying Theorem 5.2, we obtain
20
u(~)
._
<
lim inf
G(n,sla)
n~oo,...,,...,
-<
lim sup
G(n,sla)
,...,
n~oo,...,
k
Since N is arbitrary and s£[O,l)',
the theorem is proved.
,...,
CHAPTER II
EXTINCTION PROBABILITIES FOR POPULATIONS
CONSISTING INITIALLY OF ONE PARTICLE
6.
Extinction probabilities
In this chapter we will consider the probability of
extinction of the MBPRE given that Iz(O) I = 1.
~
We will
begin by introducing extinction probabilities and by
proving a result for the almost sure extinction of the
process which is similar to the well-known theorem for the
mUltitype Galton-Watson process.
Then, since the proofs
of the theorems in the later sections require some
knowledge of the dual process, we will define and study
this process.
Also, we will review some results on
products of random matrices by Furstenberg and Kesten [5].
Following these preliminaries, we will then give the best
conditions we could obtain for extinction and nonextinction
of the MBPRE given that I~(O)
I=
1.
Further, to make our
results more applicable for practical work, we will give
some corollaries whose conditions are easier to verify than
the conditions of the main results.
We begin by defining extinction probabilities.
22
Defini tion 6.1.
bility of the
~IDPRE
Let q (e.)
be the extinction -proba",,1.
given that
g(e.) = Pr{Z(n) = 0
-1.
-
-
~(O)
=
~i'
i=l(l)k; that is
for some n=l(l)-!Z(O) = e.}
--1.
-
for i=l(l)k.
Let q denote the vector
-
Since 0 is an absorbing state, the
Remark 6.1.
extinction probability may be calculated as follows:
q(e.} = Pr{Z(n) = 0
-1.
-
for some n=l(l)-lz(O) = e.}
-
00
--1.
-
-- - -
= pr{um=l [Z (m) = OlIZ(O} = e. }
-1.
= limn-+oo pr{Um~l [Z (m) = OlIZ(O) = e. }
-1.
-
= limn-+oo Pr{Z(n) = OIZ(O} = e. }
-1.
""
i=l(l)k.
= lim
G(n,Ole.},
n-+oo
- ""1.
From these calculations and the fact that 0 is absorbing,
""
i t is clear that the extinction probability q(e.) is
""1.
precisely the probability that the MBPRE ever becomes
~(O)
extinct, given that
Theorem 6.1.
= e .•
""1.
Suppose the MBPRE is impartial.
If for
some integer i, l<i<k, q(e.) = 1, then q = 1.
-
-
""1.
- - - - ""
""
Assume q(e.} = 1 for some integer i, l<i<k.
Proof.
-1.
Since the MBPRE is impartial, there exists a positive integer
N such that
MN
""
is positive (see Definition 4.3).
G(n,ole.} = L
"" -1.
Now, if
~£T
a£
""
T Pr{Z(N)
""
For n>N,
= alz(O) = e.} G(n-N,Ola).
- ""
-1.
- -
and a(j}>O for some integer j, 12j2k, then
23
Remark 3.2 implies that G(n-N,Ola) < G(n-N,Ole.) for n>N.
~
~
~
~J
Thus for j=l(l)k and n>N,
G(n,ole.) < Pr{Z(j,N) = Olz(O) = e.} +
-
-1
-
--1
Pr{Z(j,N) > OI~(O) = e.} G(n-N,Ole.).
-1
Taking limits, as n
+
-
-J
00, we obtain for j=l(l)k,
Since ~ is positive, pr{Z(j,N) > OI~(O) = e.} is positive
-1
for j=l(l)k.
-=
q
Thus, q(£j) = 1 for j=l(l)k.
Equivalently,
1.
Before defining the dual process, let us prove a
result for the almost sure extinction of the MBPRE.
Observe
that the method of proof is essentially the same method
used in proving a similar result for the multi type
Galton-Watson process (Harris [8, p. 41]).
Theorem 6.2.
Suppose the MBPRE is insecure, impartial,
and has finite moment
Proof.
-
matrix~.
If
-
A(~)
-
< 1, then q = 1.
----
Suppose the MBPRE is insecure, impartial, and
has finite moment matrix
some integer i,
l~i~k.
-
~.
Further, suppose Z(O) = e. for
'"
"'1
Now since the process is insecure,
(5.3) of Theorem 5.2 implies that
pr{IZ(n)
'"
1+
-
O} + Pr{lz(n)
Also, Remark 4.1 implies that
~
-
1+
oo} = 1-
is primitive and hence
-
A(~)
24
is well-defined.
Thus, combining Corollary 3.2.1,
Theorem 4.1, and the assumption that
-I
-
A(~)
< 1, we can infer
-
that E{IZ(n) I} is a bounded function of n.
-
Pr{IZ(n)
It follows that
-
~ oo} = 0 and therefore that Pr{IZ(n)
Thus, q(e.) = 1 for i=l(l)k.
I
~ O} = 1.
-1
Remark 6.2.
We note that the transience of the nonzero
states is used to obtain this result; in- particular, we
have used Theorem 5.2.
Further, if
e
contains only one point, then Theorem 6.2
is similar to the standard result for almost certain
extinction of the multi type Galton-Watson process.
7.
The dual process
Definition 7.1.
Consider the discrete parameter
vector Markov process {X(n),
n=O(l)-} taking values in
....,
[O,l]k and defined as follows:
~E:
[0,1]
k
for arbitrary, but fixed
,
~(O) =
E'
-
X(n+l)
n=O(l)-.
By Remark 2.4, {X(n),
n=O(l)-} is indeed a family of well,...
defined random vectors.
{~(n),
The stochastic process
n=O(l)-} will be called the dual process associated
wi th the MBPRE.
Remark 7.1.
Note that the dual process is not unique.
25
In fact, it would be better to say that
{~(n),
n=O(l}-} is
the dual process associated with the MBPRE and the vector
~£[O,l]
k
•
For convenience however, we will call each
process the dual process.
Remark 7.2.
A direct consequence of Definition 7.1
is that
Since for any nonnegative integer n, ,...
s£[O,l]k, and
... ,
en }
£
en + l
'
a vector of m.p.g.f., it follows, with probability one, that
(7.2)
,e(n} 1,e(0} =
Remark 7.3.
~£ [O,l]k
>
X(n}
IX(O}
= 0,
,..
,...
,...
n=O(l}-.
For the expected location of the dual
process after n steps, we obtain straight forwardly that
whenever ,...
s£[O,l]
k
•
Combining this with (3.3) of Remark 3.2,
we have
(7.3)
E{X(j,n} IX(O} = s} = G(n,sle.}
f"tJJ
""'J
I"tJ
f"tJ
for s£[O,l]k,
j=l(l}k, and n=O(l}-.
,..
8.
Products of random matrices
At this point we introduce some results on products of
random matrices.
We will begin by stating some very elegant
results by Furstenberg and Kesten [5] on the norms of
26
products of random matrices.
Since we will be mainly
interested in the row sums of products of random matrices
however, we will then introduce a class of matrices in
which the results of Furstenberg and Kesten hold for the
row sums of products of random matrices.
After some
corollaries and remarks, we take up the main question of
conditions for extinction and nonextinction of the MBPRE.
If x is any positive number, let
Notation.
log+ x = max(O, log x) •
Theorem 8.1.
Suppose
{~n'
n=l(l)-} is an infinite
sequence of independent and identically distributed hxh
random matrices.
If E{log+
I I~ll I}
<
then
00,
(8.1)
exists and
< r <
-00
00.
Further,
(8.2)
almost surely.
Proof.
This result follows from the work done by
Furstenberg and Kesten [5, pp. 459-462] and Theorem 1.2 of
Doob [3, p. 460].
A very clear and concise proof of part of
this result is also given by Grenander [7, p. 161].
Remark 8.1.
Observe that if A and Bare hxh matrices
"'"
'"
with real components, then II~ ~II ~ 111J11·11~11·
27
Corollary 8.1.1.
{~,
Suppose
n=l(l)-} is an infinite
sequence of independent and identically distributed hxh
random matrices.
If E{log+
I IMII I}
<
00,
< r
<
00.
defined in (8.1), exists and
Proof.
-00
then r, ~
Further,
Let m and n be any positive integers.
Using
Remark 8.1, we obtain
1 og
I 1M···
~1-2
~ II
almost surely.
E~-Ol
1= 1 og 11M
-ni+l ···M
-ni+n I I '
<
Thus, since
{~,
n=l(l)-} is an infinite
sequence of independent and identically distributed random
arrays,
Using (8.1), we can now verify (8.3).
Since, in later sections, we will be more concerned
with the row sums, instead of the norms, of products of
random matrices, let us introduce a special class of
We will show that if {M , n=l(l)-} takes
-n
only values in M(C,N) and the hypotheses of Theorem 8.1 are
matrices M(C,N) •
satisfied, then Theorem 8.1 will remain valid if we replace
the norm in (8.2) by any particular row sum.
Definition 8.1.
For C
~
1 and N a positive integer,
let M(C,N) be a class of hxh nonnegative, finite matrices
such that if ~i
£
M(C,N)
for i=l(l)N and ~ = ~1~2···~'
28
then
a)
and b)
....A is positive
[max;
1 <
-
Example 8.1.
. A ( i , j) ] / [min.
~,J
. A ( i , j)] < C <
~,J
00.
The following four matrices form a class
of the type M(3,2) •
011
o
1
1
1
1
1
1
1
1
1
1
1
1
o
110
1
1
o
1
1
1
0
Observe that once we verify condition (a) of Definition 8.1,
we automatically know there is a C
~
1 satisfying condition
(b) of Definition 8.1 since there are only a finite number
of 2-fold products.
Of course, if we require a value for
C, we must make further calculations.
Thus, to demonstrate
that any finite class of hxh nonnegative, finite matrices
is of the form M(C,N), we need only verify condition (a)
of Definition 8.1.
Lemma 8.1.
and A = ....1
AA
--_....
....2
If hi £ M(C,N) for i=l(l)n, where n
N,
···A
~ is positive and
.... n' then .
_-
(8.4)
<
Proof.
~
If ....
B
£
e.Al
....
J
<
II~.II,
.-
j=l(l)h.
M(C,N), then BN is a positive matrix •
This implies that each row and each column of ,...
B contains
nonzero elements.
Since, by Definition 8.1,
~1~2···~
is
positive, A
,... must be a positive matrix also.
Since the second inequality of (8.4) is obvious, we
29
we need only verify the first inequality of (8.4).
if D
,.."
= A"'1N".l
A_··· A.._
and
N~
e.Al / e.Al
'"1. "''''
'" J '" '"
< (max
Indeed,
(Em~_l D(i,m) e Bl)/(E hI D(j,m) e Dl)
=
'" m"'-
r,s D(r,s»
for i,j = l(l)h.
-m --
m=
h
e Bl)/(min r,s D(r,s»
(E __
m l -m",-
O:m~_'11 -m",,,,
e Bl)
Since A is positive, E h e Bl is positive
'"
m=l "'m"'-
and therefore
e, A 1 /
Hence, C
-1
A 1
'"
e, A 1
""'J""''''''
<
C,
i,j = l(l)h.
"""1
,."",,...,
<
e, A 1 for i,j = l(l)h and moreover,
-J - j=l(l)h.
Corollary 8.1.2.
Suppose {~n' n=l(l)-} is an infinite
sequence of independent and identically distributed hXh
random matrices which takes only values in some space M(C,N) •
If E {log +
I I!:h I I}
<
00,
then
r = I'l.mn-+oo n -1 E { log
(8.5)
exists and
-00
< r <
00.
11M
M ••• ",n
M II}
-1-2
Further,
r = I'
n- l log(e. M M ···M 1),
1.mn -+00
'" J
-1-2
-n-
(8.6)
almost surely, for j=l(l)h.
Proof.
This result is a direct consequence of Theorem
8.1 and Lemma 8.1.
Remark 8.2.
Although these results seem to be strong
and informative, they have two drawbacks.
First we will be
30
more interested in row sums than norms of products of
random matrices when we consider extinction of the MBPRE.
Thus, if we could replace the norm in (8.2) of Theorem 8.1
by a particular row sum and if we could replace the
hypothesis of Theorem 8.1 that E{log+
E{lOg+ ~J~
e.Mll}
<
~
00
I I~ll I}
<
00
by
for some integer j, then Theorem 8.1
would be more useful to us.
Now, however, to use the
above results effectively, we will have to add conditions
to our theorems on extinction of the MBPRE which will limit
their generality.
Secondly, and probably most important,
r, as defined in (8.1), appears to be exceedingly difficult
to calculate.
This is most unfortunate, since we shall
soon see that whether r < 0 or r > 0 determines, under
suitable conditions, whether or not the probability of
extinction of the MBPRE is 1.
Certainly, Corollary 8.1.1
does provide an upper bound for r which in some cases will
allow us to conclude that r < 0, but we have no similar
result from which we will be able to conclude that r > O.
Therefore, if we are to make the results of this section
more applicable to the MBPRE,we must find a simpler method
for evaluating r, and we must consider the relation
between the row sums and the norms of products of random
arrays more carefully.
9.
A condition for almost certain extinction
We will now state and prove the strongest result we
31
have been able to obtain for almost certain extinction of
the MBPRE,given that I~(O)
I
= 1.
To make this result
easier to use, we will then give some corollaries which
involve simpler, albeit less general, conditions.
After
these corollaries, we will combine the results on products
of random matrices with Theorem 9.1 to obtain a less
general, but more informative result for almost certain
extinction of the MBPRE.
From this result we will then
obtain a corollary which is possibly the most useful
corollary we shall give.
We will conclude this section
with some remarks on extensions of our results and state
an important conjecture.
Theorem 9.1.
Suppose the MBPRE is impartial.
Suppose,
also, that for some integer jo' l::.jO::.k, and some number r,
-00
< r
(9.1)
< 0,
r
> lim sup
n+oo
(n+l)-l log[e,
..::a;.:::l:.:.:m~o:...;;s;;...t;:;....s;;.u=r.;;e.;;lAy...:·:.-._T;;.h~e~n ~
Proof.
'"
=
§.(r;;0)~.(r;;1) "·§.(r;;n)
1],
.'"
J 0 ._.-
!.
Assume that for some integer jo' l::.jO::.k, and
some number r,
-00
~
r < 0, (9.1) is true almost surely.
Now, for fixed s£[O,l]k, Assumptions A2 and A4 and
'"
Jensen's inequality imply that for i=l(l)k and n=O(l)-,
(9.2)
almost surely.
<j>(i,r;; ,s)
n '"
>
n.
t,; (i, j , r;;n)
k
J=
1 s(J')
,
For notational convenience, let
defined for 2£[0,l]k by
~(r;;
'"
,s) be
n '"
32
=
1jJ(i,l; ,s)
n ""
for i=l(l)k and n=O(l)-.
Consider now the infinite sequence of random vectors
{X(n), n=O(l)-} which takes values in (O,l)k, almost surely,
and is defined by
(9.3)
If Y (n)
""
=
Y(O)
""
=
tl, for some arbitrary, but fixed t£(O,l),
""
n=O(l)-.
... ,
[Y (1, n) ,
Y(k,n)) for n=O(l)-, observe that
(9.4)
for j=l(l)k and n=O(l)-, and that, indeed Y(j,n) is a
random variable.
Furthermore, combining Assumption A2,
(9.2)
and (9.3), we obtain
(9.5)
E{~(n) I~(O)
=
t,!}
= E{t(l;n-l'
t(l;n-2' t(l;n-3'
~ E{t(l;n-l' t(l;n-2' t(l;n-3'
< E{!(l;n_l'
t(l;n-2' t(l;n-3'
· .. , t (sO'
· .. , t (l;0'
· .. , t(l;o'
tl)
••• ) }
,.,
tl)
,., .•• ) }
tl) .•• ) }
""
< •••
2.
E{t(sn-l' t(sn-2' t(sn-3'
= E{~(n)
1*-(0)
=
t,!},
••• , t(l;O' t~} ••. )}
n=O(l)-.
Let us now choose £ so that t < 1-£ < 1 and write,
for convenience,
33
for j=l(l)k and n=O(l)-.
Then there exists c > 0 such that
log[{log(l-£)}/log t] = -c.
Also, using Remark 2.6, we
observe that ~j ~(~n)~('n-l) ···f(~o) land S(j,n) are
identically distributed for j=l(l)k and n=O(l)-.
(9.6)
Thus,
pr{Y(jo' n+l) < l-£} = Pr{13 (jo,n) > [log (l-e:) ]/log t}
= Pr{ (n+l)
for n=O(l)-.
-1
log f3 (jo,n)
> -c/(n+l)}
Thus, for sufficiently large n, since r < 0,
(9.7)
But, by assumption, lim sup n-+oo (n+l)
-1
log f3 (jo ,n) < r < r/2,
almost surely; so we conclude
pr{Y(jo' n+l) < l-£}
Therefore, Y(jo,n)
+
as n
0,
+
1, in probability, as n
+
+
00.
Since
00.
Y(jo,n) is bounded by 1, the dominated convergence theorem
implies that E{Y(jo,n) I!(O)}
+
1, as n
+
00.
Combining
this with (9.5) and (7.3), we have that G(n, tIle. )
+
- -JO
as n
+
00.
1,
Since t < 1 and
G(n, tIle. ) < G(n,Ole. ) + t[l - G(n,Ole. )]
- -J 0 - -J 0
- - J0
for n=O(l)-, it follows that G(n,Ole. )
- -JO
+
1, as n
+
00.
Thus
q(e. ) = 1 and by Theorem 6.1, q = 1.
-)0
Remark 9.1.
The application of Jensen's inequality in
the proof of Theorem 9.1 should be noted.
While studying
Smith and Wilkinson's [11, p. 820] method of linear
34
approximation, we were tempted to use hyperplanes to bound
the m.p.g.f.
~(i'~n'~)
in (9.2).
Although this approach
might have led to a sharper result (i.e., r
~
0), we found
that the necessary approximations were difficult to handle,
since they could not be placed in closed form.
If, on the other hand, we apply the Jensen's inequality
approach to the BPRE and observe that the right hand side
of condition (9.1) is a generalization of the BPRE
condition for almost certain extinction, we obtain the BPRE
result for r < O.
Smith and Wilkinson have a sharper result,
however, which is apparently due to their use of linear
approximations in the case r
Remark 9.2.
Suppose the MBPRE is insecure, impartial,
and has finite moment
matrix~.
Theorem 6.2 implies that q
""
consequence of Theorem 9.1.
A=
A(~)
If
A(~)
< 1, then
This result is also a
To demonstrate this, let
If r
=
log Al ,
Moreover, using the Markov lemma, we obtain
<
~
=
""
1.
and choose A such that A < A < 1.
l
l
then r < O.
Since
= O.
e
c n +l 1
-n-l
",,1 ~
"" Al
'
n=O(l)-.
is primitive, Theorem 4.1 implies that there exists
positive integers K and L such that
35
n>L.
Thus,
n>L.
Since En : L K(A/Al}n+l is finite, the Borel-Cantelli lemma
implies that there exists a positive random variable, finite
almost surely, say N(w}, such that if n
Hence condition (9.1)
r = log A < O.
l
hold~
~
N(w},
almost surely, for jo = 1 and
Theorem 9.1 can now be applied to prove
that q = 1""
""
Note that this result is weaker in one respect and
stronger in another, than Theorem 6.2.
On the one hand, we
have made no use of the fact that the MBPRE is insecure.
On the other hand, we have proved this result only for
A(t} < 1 and not for A(t} < 1.
""
""
Corollary 9.1.1.
Suppose the MBPRE is impartial and
matrix~.
has finite moment
""
Corollary 9.1.2.
Further, suppose
n=O(l}-.
Then,
~(s
n
If A(t} < 1, then q = 1.
-
--,..,
""
""
Suppose the MBPRE is impartial.
) = max{e.
~(s
""J ""
{~(sn)'
n
) 1, j=l(l}k} for
""
n=O(l}-} is an infinite sequence of
independent and identically distributed random variables.
If E{log+ ~(sO}} <
00
and E{log ~(sO}} < 0, then ~ = 1.
36
Proof.
If
~(I';
n
) = max{e.
~(I';
-J -
n
) 1,
-
n=O(l)-, then Remark 2.6 implies that
j=l(l)k} for
{~(I';
n
), n=O(l)-} is
an infinite sequence of independent and identically
distributed random variables.
~(I';n)'
{log
Moreover,
n=O(l)-} is an infinite sequence of
independent and identically distributed (possibly extended)
Suppose E{log+
random variables.
E{log
~(I';O)}
< O.
~(I';O)}
<
00
and
Then, with probability one,
n=O(l)-.
It follows, using the strong law of large numbers, that
lim sup n ~
almost surely.
Since this is condition (9.1), Theorem 9.1
implies that q = 1.
'"
Corollary 9.1.3.
Suppose the MBPRE is impartial.
Further, suppose there is an infinite sequence of
independent and identically distributed nonnegative random
matrices {0(1';
'" n ), n=O(l)-} such that _~(I'; n ) -< _0(1'; n ), with
---probability one, for n=O(l)-, and that for some integer
jO.' l.::.jO.::.k, and some number r,
-00
r > lim sup ~ (n+l) -1 log [e.
n~
~
r < 0,
0 (1';0) 0 (1';1)···0 (I'; ) 1],
n -
-JO -
_a_l_m_o_s_t_s_u_r_e_l....y_ _•__T_h_e_n
~
~.
{~(I';
), n=O(l)-} is an infinite
'" n
sequence of independent and identically distributed random
Remark 9.3.
Since
=
37
matrices, it follows that Pr{I~(s
)
..... n
of n.
Note that, if pr{lf(so)
I
I
= O} is independent
= O} is positive, then there
would be a positive probability of extinction at each
generation.
In this case, q = 1.
Corollary 9.1.4.
Suppose the
r~PRE
is impartial.
Suppose, also, that for some positive vector
y
= (v(l) , ••• ,v(k)), there is an infinite sequence of
positive random variables {A(n), n=O(l)-} such that
~(s
n ) v = A(n) v, almost surel~ for n=O(l)-. If
E{log+ A(O)} < 00 and E{log A(O)} < 0, then q = 1.
""J
I"tJ
,....,
--
Proof.
Since A(n) =
---
'"
Lj~l ~(l,j,sn) v(j)/v(l), almost
surely, for n=O(l)-, i t follows, from Remark 2.6, that
{A(n), n=O(l)-} is an infinite sequence of independent and
identically distributed random variables.
Observe, that
if E{log+ A(O)} is finite', then {log A(n), n=O(l)-} is an
infinite sequence of independent and identically distributed
random variables whose first moments are well-defined.
Assume now that E{log+ A(O)} <
00
and E{log A(O)} < O.
If we let u = min{v(i), i=l(l)k} > 0, then, with
probability one,
=
n=O(l)-.
It follows, with probability one, that
38
for n=O(l}-.
~
Taking limits, as n
00, and applying the
strong law of large numbers, we obtain, with probability
one,
Theorem 9.1 now implies that .....
q = .....
1.
Remark 9.4.
Corollary 9.1.4 is particularly useful
if for each environmental state
a,
the mean number of
particles generated by any particle is the same for all
types.
In this case, v..... = 1.
Lemma 9.1.
Let M denote a space of primitive hxh
nonnegative matrices which commute under multiplication.
Then there is a positive vector v such tilat M
v = A(M} v
..........
for each M
£
Proof.
M.
Choose A,B
~
~
M.
£
Let A(A} and A(B} denote
~
~
the corresponding largest positive characteristic root of
b
and~.
Denote by
~(~)
and
~(~)
the right, positive,
normalized eigenvector associated with A(A}
..... and A(B},
.....
respectively.
Then A v(A} = A(A} v(A} and
B
= A(B}
v(B}.
,...., v(B}
,....,
,....,,....,
,....,
,....,
t"tJ
,....,
,....,
".",
t"tJ,....,
Recall that the right eigenspaces
associated with A(h} and A(~} are one-dimensional and
that v(A}
,..., and v(B}
'" ,...., are the only positive, normalized
,..."
right eigenvectors associated with A and .....
B, respectively .
A w = cw
Thus, if ..........
..... ' where w is a positive vector and c is a
39
nonzero constant, then
~
is a mUltiple of
~(h).
Of course,
~.
a similar result holds for
Now since
B A v(B)
"J
"J
=
"J
=
=
A B v (B)
"J
and since A v(B) is positive, it follows that A v(B)
~~
,-v",
for some nonzero constant c.
~(~)
= ~(~).
Thus, since
a positive vector
~
h
=
cv(B)
""t"W
Since v(B) is positive,
"J
£
such that
"J
M is arbitrary, there exists
~(~)
=
~
~ £
for every
M.
The result now follows.
Corollary 9.1.5.
Suppose the
l~PRE
is impartial and
suppose that {;(~ ), n=O(l)-} takes values in a space of
"J
n
primitive kxk matrices, which are commutative under
multiplication •. Then there exists a positive vector v and
an infinite sequence of positive random variables
{A(n), n=O(l)-} such that, almost surely, ;(~ ) v = A(n) v
-
for n=O(l)-.
If, in addition, E{log+ A(O)} <
E{log A(O)} < 0, then q
- -
Proof.
n
"J
"J
=
00
and
1.
"J
This result follows directly from Lemma 9.1
and Corollary 9.1.4.
Clearly, the primary difficulty in applying Theorem 9.1
is the verification of condition (9.1).
willing to assume that E{log+
I I~(~o) II}
However, if we are
<
00,
then we can
gain greater insight into condition (9.1) by applying the
results on products of random matrices to the MBPRE.
Hence,
40
let us assume that E{log+
I If(so) I I}
<
00.
We can now present a result which is extremely
informative, but unfortunately not as sharp as Theorem 9.1.
This drawback is small, however, in view of the very
applicable corollary which is a direct consequence of this
theorem.
Theorem 9.2.
E{ log+
I If
( sO)
Suppose the MBPRE is ·impartial and
I I}
<
00.
Then
< r
<
00.
Further,
(9.8)
exists and
-00
(9.9)
:::a=l:::m=o.::.s.:t--=.s.;u;:;:,r,;::;e,;:,ly
........_=I=.f r < 0, then q = 1.
- - '"
Proof.
Combining the results of Theorem 8.1 with
Remark 2.6, we obtain (9.8) and (9.9).
Also, since
<
almost surely, for n=O(l)-, it follows that
lim sup n-+oo
almost surely.
Thus, if r < 0, then Theorem 9.1 implies
that q = l .
'"
Corollary 9.2.1.
E{log+
I If(so) I I}
<
00.
Suppose the MBPRE is impartial and
Suppose also, that there is a
41
positive integer N such that
(9.10 )
Then q = 1.
Proof.
This is a direct consequence of Corollary 8.1.1
and Theorem 9.2.
Remark 9.5.
for r.
Observe that (9.8) gives an expression
If we could evaluate this limit in (9.8) to find r
exactly however, we could probably, just as easily, evaluate
the limit superior in condition (9.1) or the limit in
condition (9.9).
On the other hand, Corollary 9.2.1 does provide us
with conditions which might be easily verified in a real
situation.
If we were extremely fortunate and could verify
(9.10) for some positive integer N, then Corollary 9.2.1
would be the most important result stated so far.
However,
if we could not find an appropriate N to satisfy (9.10) and
we could not infer that {E 10g[llf(so) ···f(sn)
Ill,
n=O(l)-}
was bounded below by some positive number, ther. we could
not conclude that r < 0 or r > O.
In this case, the above
results would have little significance.
We now conclude this section on almost certain
extinction of the MBPRE, given that I~(O)
conjecture.
NBPRE.
I
= 1, with a
First, however, we need to define a singular
42
Definition 9.1.
The MBPRE is said to be singular if
the multivariate probability generating functions
G{l,sle.), j=l{l)k, are all linear in s{i), i=l{l)k, with no
'" "'J
constant terms; that is, each particle generates exactly
one particle.
Remark 9.6.
If the MBPRE is singular, then
~(~
'"
n
) 1 = 1,
almost surely, for n=O(l)- and
lim sup n-+oo
almost surely, for j=l{l)k.
It also follows that q = O.
'"
As noted in Remark 9.1, Smith and Wilkinson were able
to prove, for the BPRE, a result similar to Theorem 9.1 for
r < 0 and not just r < O.
They did, however, use a result
by Chung and Fuchs [2] which appears to be crucial in
proving the result for r = o.
We believe that "similar"
results are needed for the case r = 0 of the MBPRE and
that, in view of Remark 9.6, the following conjecture is
true.
Conjecture 9.1.
nonsingular.
jo'
l~jO~k,
Suppose the MBPRE is impartial and
Suppose, also, that for some integer
and some r,
almost surely.
_00
Then q = 1.
< r < 0,
43
10.
A condition for nonextinction
In this section we shall state and prove a theorem
which gives conditions under which there is a positive
probability of survival of the MBPRE, whenever
I~(O)
I
= 1.
Before presenting this result, however, we must prove a
technical lemma.
After doing this and after presenting
our main result, we will comment on the various hypotheses.
Then, by applying the results on products of random
matrices, we will give an equivalent theorem which is more
informative, but equally difficult, we believe, to apply.
As in Section 9, we will then state some corollaries which
involve simpler, but less general, conditions.
We will
conclude with a comment on necessary and sufficient
conditions for almost certain extinction of the MBPRE, given
tha t
I z (0) I
'"
= 1-
Lemma 10.1.
Let W(l), W(2),
.•. , W(h) be nonnegative,
integral valued random variables and suppose EIW(i)
for i=l(l)h.
i=l(l)h.
I
<
00
Denote the finite mean E Wei) by m(i) for
Then for every E, 0 < E < 1, there exists a real
number teE) E [0,1) such that
(10.1)
<
IT . h 1
J=
( .) Em ( j )
s J
,
whenever.e E [teE), l]h.
Proof.
Assume h=l.
If mel) = 0, then W(l) = 0, almost
surely, and hence (10.1) will be satisfied for every
44
E, O<E<l, if t(E) =
o.
Suppose then that m(l)
for some E, O<E<l, there does not exist t(E)
that (10.1) is satisfied.
sequence t
and
l
> 0 and that
E [0,1) such
Then there exists an increasing
,t ,t , ... of real numbers such that
2 3
i)
t
ii)
t
iii)
t
E (0,1)
n
1,
-+
n
Em(l)
n
for n=l(l)-,
as n
-+
00,
< E{t W(l)}
n
'
n=l(l)-.
Thus, using (iii), we obtain for n=l(l)(1 - t Em(l))/(l - t )
n
n
which implies, as n
-+
>
(1 - E{t W(l) })/(l - t
n
00, that Em(l)
~
m(l).
)
'
Since m(l)
is positive and O<E<l, this is a contradiction.
there exists t(E) e
n
Hence,
[0,1) which satisfies (10.1) for h=l.
Assume now that h is an integer greater than one.
By
repeated application of the Schwarz inequality, we have
E{II,h s(),)W(j)}
)=1
2. [II~:i
for 2E[0,1]
h
•
j
-j
(E{S(j)2 W(j)})2
]
h-L
(E{s(h)2
Choose E such that O<E<l.
for h=l, there exist
real numbers t(i,E)
I-h
w(h)})2
,
Since (10.1) holds
E [0,1) for i=l(l)h
such that E{s(i)W(i)} < s(i)Em(i), whenever t(i,E) ~ s(i)
for i=l(l)h.
< 1,
Combining these results, we obtain inequality
2i
(10.1), whenever t(i,E) < s(i)
< 1 for i=l(l)h-l and
h-l
t(h,E) < s(h)2
< 1.
Thus, if we define t(E) by
-i
t(E) = max{t(i,E)2
, i=l(l)h},
45
then t(£) £ [0,1) and (10.1) is satisfied whenever
~
h
£ [t(£) ,1] .
Theorem 10.1.
Suppose the space 8 of the MBPRE
contains only a finite number of points.
Further, suppose
that {f(sn)' n=O(l)-} takes only values in some space M(C,N)
and that for some integer jo'
l~jo~k,
and some real number
r > 0,
(10.2)
r < lim inf n ~ (n+l)-l log[e.
~(sO)~(sl)·"~(s) 1],
""'JO "'"
with probability one.
"'"
n
"'"
Then q < 1.
Proof.
We consider two cases.
Case 1.
(N = 1)
£ = exp(-r/2).
"'"
We begin by choosing £ such that
Clearly 0<£<1.
Applying Lemma 10.1, we
have for every 8£8 there exists a real number t(8) in
(0,1) such that
~(J' 8 s) <
(10.3)
'Y
,
,
IV
-
k s(i)£~(j,i,8)
rr.1.=1
for j=l(l)k, whenever s £ [t(8), l]k.
"'"
Since 8 contains only
a finite number of points, if t = max{t(8), 8£8}, then
t£(O,l) and (10.3) is satisfied for j=l(l)k and 8£8,
k
whenever s£[t,l] •
"'"
Now for any s£[t,l]k there is a unique nonnegative
IV
vector w such that w£[O,l]k and s = (tw(l) ,tw (2) , ..• ,tw(k».
IV
"'"
For j=l(l)k, 8£8, s£[t,l]
"'"
let us define
~(j,8,s)
IV
by
k
w(l)
w(k)
and s = (t
, ... ,t
),
"'"
46
(10.4)
=
r
",('
8 ,~ ) -- ,1,('
8,
't' J,
't' J,
Xp {(£
L:i~l
(tw(l) , t w (2) , ..• , tw(k»)
w(i) s(j,i,8»
~XP{(£m-l L:i~l w(i)
log t},
s(j,i,8» log t},
if m < 1,
if m > 1,
where
k
m = max{£ L: i = 1 w ( i ) s ( j , i , 8), j = 1 ( 1) k } •
Observe that 1jJ(j,8,s)
£ [t,l] and that by (10.3)
,...
( 10 .5)
< 1jJ(j,8,s)
""
for j=l(l)k, 8£8, and ,...
s£[t,l] k .
Consider now the infinte sequence of random vectors
{Y(n)
= [Y(l,n), •.• ,Y(k,n)], n=O(l)-} which takes values in
,...
[t,l]k and is defined by
Y(O)
=
tl,
Y(n+l) = t(~n' Y(n»,
n=O(l)-.
It follows that, for j=l(l)k,
Y(j,l) = exp{ (£~j ~(~O) ~/max[l, II£f(1';o) II]) log t}
and Y (j, 2) =
t£2~j ~(~l)f(~o)
l-/max[l,
For convenience, let us write
11£~(~o) II, 11£2f(1';1)f(~0) II]
47
p(j,n)
TI(j,n) =
e:
n+l
~j ~(I;;n) f(l;;n-l)·· ·f(l;;o) ~
lI(j,n) = max[l;
IIe: i + l ~(l;;i)f(l;;i-l)·· ·f(l;;o) II,
for n=O(l)- and j=l(l)k.
i=O(l)n]
Then, in general, we have
Y(j, n+l) = exp{ [TI (j ,n) /ll (j ,n)] log t}
for n=O(l)- and j=l(l)k.
Clearly Y(j,n) is a random
variable, and by Assumption A2 and (10.5) we obtain
(10.6)
E{x,(n) 1x,(0) = tl}
,...
= E{!,(l;;n-l' !,(l;;n-2' t(l;;n-3'
> E{!(l;;n-l' ~(l;;n-2' ~(l;;n-3'
> E{!(l;;n-l' !(l;;n-2' !(l;;n-3'
·.. ,
·.. ,
· .. ,
t (1;;0' tl) ••• ) }
~ (1;;0' tl) ••• ) }
••• ) }
! (1;;0' tl)
,...
>
> E{2(l;;n-l' ;£(l;;n-2' 2(l;;n-3'
... , 2(1;;0' tl) .. ·)}
= E{~(n) I ~(O) = tl),
n=O (1) -.
Observe now that (10.2) implies that
lim infn+oo (n+l)
almost surely.
-1
log p(jo,n)
~
r/2,
Hence for small positive 8, 0<8<1, there
exists a positive integer nO such that
Pr{log p(jo,n)
~
0,
n=no(l)-} > 1 - 8/2.
Further, there exists a sufficiently large positive
such that
X
o
48
This implies that
Pr{log p{jo,n) ~ -x O '
(lO.?)
o.
n=O{l)-} > 1 -
-1
To conclude this proof for N = 1, let a = C
exp(-x ).
O
Then for n=O{l)-,
(10.8)
Note that if A = A A ···A
and 1'1,.....,
B = B B ···B
I"oJU
l 2
l 2
-
#"ttl
~
,...,
AI' ••• , A , Bl , ••• , B
~
~
~
~v
I"V
where
E: M (C, 1), then by (8. 4)
~v
(~j ~ ~ 1)/\ I~I I
=
(Ei~l
-1
> C
A{j,i)
~i ~ ....1) / II B II
~
e. A 1,
"'J ,...,
j=l{l)k.
I"V
Combining this fact with the fact that aC < 1, we obtain,
for n=O{l)-,
(10.9)
'
n+l) < t a }
P r {y( JO'
> Pr{min[l; E:n+l-ie.
-
~J
0
~{Z;; )~(Z;;
~
n....
n-
l)···~{Z;;·) 1, i=O{l)n] > aC}
....
~
~
-
This last equality follows by renumbering the independent
and identically distributed random matrices; in particular,
we substitute
~(Z;;m)
using (lO.?), that
for
~(Z;;n-m)'
m=O{l)n.
It now follows,
49
(10.10)
> 1 -
8,
n=O(l)-.
Since 0 < a < 1 and tc(O,l), we conclude that
1 -
(l-t a ) (1-8)
<
Hence,
lim sUPn~ E{Y(jo,n) I~(O) = tl} < 1.
(10.6) ,
lim sUPn~~ E{X(jo,n) I~(O) = tl} < 1.
(7 • 2) of Remark 7.2 that lim sup ~
n~oo
-
1,
n=O(l)-.
Further, by
It follows by
__
E {X (j 0' n) I X(0)
= O} < 1.
Thus, combining (7.3) and Remark 6.1, we can infer that
q (e. ) < l.
-JO
Now since
{~(s
-
n
), n=O(l)-} takes only values in some
space M(C,l), Definition 8.1 implies the MBPRE is impartial.
-
Thus, Theorem 6.1 implies that q < 1.
Case 2.
(N > 1)
In Remark 3.3 we verified that the
-
process {Z(nN), n=O(l)-} is a MBPRE.
It follows immediately
from the hypotheses of this theorem that eN contains only
a finite number of points, that
takes only values in some space M(C,l), and that
r
< lim inf
almost surely, where
i=O(l)-.
~
n~oo
(n+l)-llog[e. TOT1···T 1],
- J0 - -n-
li = f(siN)f(~iN+l) ···~(~iN+N-l)
for
Thus, we may apply the results of Case 1 to the
50
-
process {Z(nN), n=O(l)-}.
lim
G(nN,Oje.) < 1
-J
implies that q < 1.
n+oo
Remark 10.1.
In particular, we have
for j=l(l)k.
Remark 6.1 now
It might be helpful to give some comments
on the hypotheses of Theorem 10.1.
To
begi~,
by assuming that
_
n
{~(s
), n=O(l)-} takes
only values in some space M(C,N), we not' only guarantee
that the MBPRE is impartial, but we also make it possible
to continue the proof from (10.8) to (10.9).
Of course,
~(j,8,s)
a more judicious choice of m in the definition of
-
(see (10.4)) could possibly have eliminated the use of the
norm.
However, we need to be sure that
-
~(j,8,s)
always
takes values in [t,l] so that the Y(n) process is well-
-
defined and so that we may verify inequality (10.6).
had let
m = £
k
~i=l
w(i)
~(j,i,8),
If we
say, then the same effect
would have been obtained and the norm would have been
removed.
But, then the formulation of Y(j,n) would have
become so complex that we would have been unable to evaluate
Thus, we assumed that {~(s
_
n ), n=O(l)-}
takes only values in some space M(C,N).
Observe also, that if
_
n
{~(s
), n=O(l)-} takes only
values in some space M(C,l), then we only need the
hypothesis that 0 contains only a finite number of points
to show that there exists some t£(O,l) so that (10.3) is
satisfied whenever j£{1,2, .•• ,k}, 8£0, and s£[t,l]
-
k
•
Thus,
in a particular case, if N = 1 and we know that there is
51
some ts(O,l)
j, 8, and
~,
for which (10.3) is true for appropriate
then it is irrelevant whether 8 consists of
finitely many points or infinitely many points.
Finally, it appears that if we wish to prove a
result similar to Theorem 10.1 for which 8 may be any set,
we must find an additional condition as was the case in
the BPRE (Smith and Wilkinson [11, p. 820] and Smitn [10]).
At present, we cannot hypothesize what this condition might
be.
Before applying the results on products of random
matrices to Theorem 10.1, let us pause to consider further
consequences of Theorem 10.1 with respect to the multitype
Galton-Watson process.
Remark 10.2.
Suppose the MBPRE is impartial and the
space 8 contains exactly one point, say 8.
and Remark 4.1, ,...,~(8)
=
~
is primitive.
By Assumption A4
Thus, if
A(~)
> 1,
the standard result for extinction of the multi type
Galton-Watson process (Harris [8, p. 41]) implies that q < 1.
This result is also a consequence of Theorem 10.1.
Indeed, since
~(8)
is primitive,
~(8)
class of matrices of the form M(C,N).
forms, by itself, a
Further, if
A(~)
> 1,
then Theorem 4.1 implies that
> 0,
with probability one.
q,..., < 1-
Applying Theorem 10.1, we have, again,
52
The primary difficulty in applying Theorem 10.1 is the
verification of (10.2).
by assuming that E{log+
Recall, however, that in Section 9,
II£(so) II}
<
we gained consider-
00,
able insight into condition (9.1) when we applied the
results on products of random matrices.
we will not need to assume that E{log+
In this section,
I I~(so) I I}
since Assumption A4 and our hypothesis that
e
is finite
contains only
a finite number of points already imply this condition.
Hence, we can apply the theorems of Section 8 to Theorem 10.1
to obtain a better result.
The one noticeable defect,
unfortunately, will be the absence of an applicable
corollary similar to Corollary 9.2.1.
Suppose the space e of the MBPRE
Theorem 10.2.
contains only a finite number of points.
that
--
{~(s ),
'"
M(C,N).
n
Furthe4 suppose
n=O(l)-} takes only values in some space
Then
exists and
-00
< r
<
00.
Further,
and
r = limn-+oo
almost surely, for j=l(l)k.
Proof.
If r > 0, then q < 1.
This result follows directly from the above
comment, Corollary 8.1.2 and Theorem 10.1.
53
We now state some corollaries which parallel those
in Section 9; their proofs are too similar to merit
repetition.
Corollary 10.1.1.
Suppose the space 0 of the MBPRE
contains only a finite number of points.
Further, suppose
that {f(~n)' n=O(l)-} takes only values in some space
M(C~N)
and that
n=O(l)-.
~(~
n
) = min {e. t,;
..... J.....
(~
n
) 1, 'j = 1 ( 1) k} for
.....
Then, {~(~n)' n=O(l)-} is an infinite sequence
of independent and identically distributed random variables.
If E{log+ ~(~O)} <
00
and E{log ~(~O)} > 0, then ~ < 1.
Corollary 10.1.2.
Suppose the space 0 of the MBPRE
contains only a finite nunber of points.
Further, suppose
that {f(~n)' n=O(l)-} takes only values in some space
M(C,N) and that there is an infinite sequence of independent
and identically distributed nonnegative random matrices
{6(~
,.,.
n
t,;(~ )
), n=O(l)-} such that
for n=O(l)-.
.....
n
> 6(~ ), almost surely,
- .....
If for some integer jo'
n
l~jo~k,
and some
real number r > 0,
(10.11)
almost surely, then ,.,.
q < 1.
Remark 10.3.
One particularly interesting choice for
the infinite sequence
{6(~
.....
n
), n=O(l)-} is defined by
6(i,j'~n) = [t,;(i'i'~n)'
if i=j,
o,
if i;ij,
54
for i,j = l(l)k and n=O(l)-.
In this case, the strong law
of large numbers and Remark 2.6 imply that the right lland
side of (10.11) is equal to E{log
surely, if Ellog ~(jo' jo'~O)
I
<
~(jo'
jo'
~O)},
almost
Thus, if for some
00
jo' l'::j02.k , E{log ~(jo' jo' ~O)} > 0, then;J < 1, whenever
the other two hypotheses are satisfied.
Corollary 10.1.3.
Suppose the space 8 of the MBPRE
contains only a finite number of points.
that
{~(~
--",n
Further, suppose
), n=O(l)-} takes only
values in some space
M(C,N)
-
and that for some positive vector
~
there is an infinite
sequence of positive random variables {A(n), n=O(l)-} such
that
f(~n)
~
= A(n)
E{log+ A(O)} <
00
~,
almost surely, for n=O(l)-.
and E{log A(O)} > 0, then q < 1.
Corollary 10.1.4.
Suppose the space 8 of the MBPRE
contains only a finite number of points.
that
--
{~(~
'"
n
If
Further suppose
), n=O(l)-} takes only values in some space
M(C,N) and that in this space multiplication is commutative.
Then there exists a positive vector v and an infinite
'"
seguence of positive random variables {A(n), n=O(l)-} such
that, with probability one,
If E{log+ A(O)} <
00
f(~n)
~
= A(n) v, for n=O(l)-.
and E{log A(O)} > 0, then q < 1.
We conclude this section with a remark about the
existence of necessary and sufficient conditions for
almost certain extinction of the MBPRE, given that
I~(O)
I
= 1.
The following result, which is a corollary
55
to Theorems 9.2 and 10.2, will form the uasis for our
remarks.
Theorem 10.3.
Suppose the space 8 of the MBPRE
contains only a finite number of points.
Further, suppose
that {s(s ), n=O(l)-} takes only values in some space
--
,...
M ( C, N).
n
-
'l'hen
(10.12)
exists and
-00
< r <
00
If r < 0, then q = 1.
If r > 0,
then q < 1.
Remark 10.4.
Suppose the
~~PRE
is nonsingular and
that the two hypotheses of Theorem 10.3 are satisfied.
Further, assume that Conjecture 9.1 is valid.
Then, if r
is defined by (10.12), we can conclude, by combining
Theorem 10.3 and Conjecture 9.1, that r
and sufficient condition for q = 1.
~
0 is a necessary
Equivalently, r
~
0
if and only if the probability of the MBPRE ever becoming
extinct, given that I~(O)
I
= 1, is one.
CHAPTER III
EXTINCTION PROBABILITIES FOR POPULATIONS
WITH ANY GIVEN INITIAL POPULATION
11.
The ergodic distribution
In Chapter II we considered conditions for almost
certain extinction and nonextinction of the MBPRE with an
initial population of only one particle.
Let us extend our
discussion to the MBPRE with a fixed, but arbitrary, initial
population.
We shall begin by extending our earlier re-
sults on extinction probabilities and on the moments of
the dual process.
We will then be able to demonstrate the
existence of a multivariate distribution function each
moment of which is precisely the probability of extinction
of the MBPRE for some given initial population.
After
this, we will show that the results of Sections 9 and 10
can be easily extended to include the MBPRE with any given
initial population.
Let us begin by extending our definition of extinction
probabilities.
Definition 11.1.
-
For a£T, let q(a) denote the
~
extinction probability of the MBPRE given that the initial
57
population
~(O)
=
~;
that is, define q(a) by
'"
q(~) = pr{~(n)
Remark 11.1.
= 0 for some n=l(l)-I~(O) = a}.
Observe that Definition 11.1 is
consistent with Definition 6.1 and that by applying the same
reasoning as was used in Remark 6.1, we can conclude that
(11. 1)
q(a) = lim
'"
n -+00
G(n,Ola),
'"
aET.
'"
Moreover, combining (11.1) with our results on the various
representations of G(n,~I~)
results on the dual process
(see Remark 3.2) and our
(see Remark 7.2), we immediately
obtain that
whenever
~ET.
Now, since the dual process takes only values
in the unit k-cube, Example (d) of Feller [4, Vol. 2, p. 244]
implies that X(n) converges in distribution to some random
'"
vector V as n + 00
Further, V takes only values in
'"
(0,1] k and
,eET.
Definition 11.2.
We shall refer to
F(~)
For XE[O,l]k,
let F(x)
= Pr{V < x}.
,....,
"""-J,....,as the ergodic distribution function
assoclated with the dual process {X(n), n=O(l)-}
'"
(and
X(O) = 0).
""
'"
Theorem 11.1.
To any MBPRE, {[(n), n=O(l)-}, there
corresponds a dual Markov process,
{~(n),
n=O(l)-}, which
58
is defined on the unit k-cube and which possesses a
limiting ergodic distribution F.
The moments of Fare
q(~),
precisely the extinction probabilities
~e:T;
that is,
(11. 2)
Theorem 11. 2.
Suppose the MBPRE is impartial.
Assume
also, that for some integer jo' 12j02k, and some number r,
< r
-00
< 0,
r > lim sup n-+oo
almost surely.
Then the ergodic distribution F is degenerate
and concentrates all its mass at 1.
Also q(a) = 1 for
'"
every ae:T.
- - = - '"
Proof.
From Theorem 9.1 we can infer that q(e.) = 1
"'~
for i=l(l)k.
Applying (11.2) of Theorem 11.1, we see
immediately that
~
=
1,
with probability one, and moreover
that q(a) = 1 for every ae:T.
'"
'"
Theorem 11.3.
Suppose the space
e
contains only a finite number of points.
of the MBPRE
Further, suppose
that {f(~n)' n=O(l)-} takes only values in some space M(C,N)
and that for some integer jo' 12j02k, and some real number
r
> 0,
almost surely.
and
q(~)
-+
0
Then F(l-) = 1,
-as
la\
'"
-+
00
q(~)
<
1 for
~
e: T-{£},
59
Proof.
As in the proof of Theorem 10.1, we consider
two cases.
Case 1.
Suppose our hypotheses are valid and that N=l.
Then the initial arguments employed in the proof of
Theorem 10.1 imply that there exists a number t£(O,l) such
that (10.3) is true whenever ,e£[t,l]k, 8£8, and l<j<k.
Further, from (10.10) we have that for each 8 > 0 there is
a sufficiently large x
o
such that
C
(11.3)
-1
exp (-x )
O
pr{Y(jo' n+l) ::.. t
IY(o)
~
= tl}
~
-> 1 -
for n=O(l)-.
From the definition of the Y(n) process
(see (10.5) -
(10.6)) and Remark 7.2 we have that
~(n) I~(O)
~
=
~
>
X(n)
~
I X(O)
'"
almost surely, whenever 2£[0,1]
integer.
8
k
= s
~
>
~(n) I~(O)
= £,
and n is a nonnegative
Combining this result with (11.3), we obtain that
-1
C
pr{X(jo' n+l) ::.. t
for n=O(l)-.
exp (-x )
O
I~(O) =
£}
~ 1 -
8
Since X(n) converges in distribution to V,
'"
~
and, in particular, V(jO) < 1, with probability one.
Now Theorem 10.2 implies that
almost surely, for every j, l::..j::..k.
Thus, in the above
remarks, we can let jo be any integer, l::..jO::..k.
So, V(j) < 1,
60
with probability one, for j=l(l)k.
I1oreover, by applying
(11.2) it follows irrunediately that q (a)
<
'"
and that q(~)
Case 2.
0
+
I~I
+
'"
00.
Suppose now that our hypotheses are satisfied
and that N > 1.
{~(nN),
as
1 for a E T-{ O}
Then Remark 3.3 implies that the process
n=O(l)-} is also a MBPRE.
For convenience, let us
denote the extinction probabilities associated with the
~(nN)
process by
with the
~(nN)
qN(~)
~ET,
the dual process associated
~(n),
and the ergodic distribu-
for
process by
tion associated with the
~(n)
process by F •
N
If we apply
the arguments of Case 2 of the proof of Theorem 10.1, we
can conclude immediately that F (!-) = 1,
N
for a E T-{O}, and qN (~) + 0 as I~I + 00
""
qN (~)
A brief
'"
review of the definition of
qN(~)
(see Definition 11.1 and
Remark 11.1) however, reveals that
Moreover, although the
~(n)
< 1
qN(~)
=
q(~)
for
~(n)
process and the
~ET.
process
are not, in general, equivalent, the respective moments
of the ergodic distributions, F
N
and F, are equal.
Since
the support of both distributions is the compact [O,l]k,
F
N
and F are completely determined by their respective
moments and, we can conclude that F
F(l-) = 1,
N
= F.
Thus
q(a) < 1 for a E T-{O}, and q(a)
'*"'J
"""'J
~,...,
+
0
as lal
,.....,
+
It should now be clear that all the results presented
in Sections 9 and 10 can be somewhat generalized.
In
particular, instead of concluding that q = 1 or q < 1,
""
00.
61
we may now infer, under the same hypotheses, that q(a)
,..,
for every
~£T
or that
q(~)
< 1 for every ~ £ T-{O}.
=
1
Further,
we may continue to apply all our corollaries, whenever
possible, to obtain tnese generalized results for the MBPRE.
We conclude this work by stating a theorem which,
together with Conjecture 9.1, suggests that r < 0 may be a
necessary and sufficient condition for almost certain
extinction of the MBPRE, under certain hypotheses.
The
veracity of this statement depends, of course, on the
probability of extinction of the MBPRE when r = O.
Unfortunately, we have been unable to draw any significant
conclusions about the probability of extinction of the
=
MBPRE when r
O.
Theorem 11.4.
Suppose the space
e
contains only a finite number of points.
of the MBPRE
Further, suppose
that {s(s ), n=O(l)-} takes only values in some space
--
,..,
M(C,N).
n
Then
r
exists and
~£T.
-00
=
limn-+oo (n+l)-l E{log
< r
<
00.
11~(so)~(sl) "·~(sn) II}
If r < 0, then
q(~)
=
1 for every
If r > 0, then q(~) < 1 for every ~ £ T-{O}.
62
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[1]
Bellman, R. (1954). Limit theorems for non-commutative operations. I.
Luke Math. J.
21 491-500.
[2]
Chung, K. L. and Fuchs, W. II. J. (1951). On the
distribution of values of sums of random variables.
Mem. Amer. Math. Soc.
6 1-6.
[3]
Doob, J. L. (1953).
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[4]
Feller, W. (1957, 1966). An Introduction to
Probability Theory and its Applications,
(2nd ed.), 2. Wiley, New York.
[5]
Furstenberg, H. and Kesten, H. (1960). Products of
random matrices.
Ann. Math. Statist.
31
457-469.
[6]
Gantmacher, F. R. (1959). Applications of the
Theory of Matrices.
Interscience, New York.
[7]
Grenander, U. (1963). Probabilities on Algebraic
Structures. Wiley, New York.
[8]
Harris, T. E. (1963). The Theory of Branching
Processes. Prentice-Hall, Englewood Cliffs.
[9]
Karlin, S. (1966). A First Course in Stochastic
Processes. Academic Press, New York.
[10]
Stochastic Processes.
Wiley,
-
1
--
Smith, W. L. (1968). Necessary conditions for almost
sure extinction of a branching process with
random environment. Ann. Math. Statist.
39
2136-2140.
--
[11]
Smith, W. L. and Wilkinson, W. E. (1969).
ing processes in random environments.
Math. Statist.
40 814-827.
On branchAnn.
-~
[12]
Wilkinson, W. E. (1967). Branching processes in
stochastic environments. University of North
Carolina Institute of Statistics Mimeo Series
No. 544.
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12. SPONSORING MILITARY ACTIVITY
Logistics and Mathematical Statistics
Branch Office of Naval Research
Washington, D.C. 20360
13. ABSTRACT
Consider the k-type Galton-Watson process. In this work, we remove the restrictive assumption that particles of the same type always divide in accordance with the
same p.g.f. Instead, we assume that at each unit of time, Nature chooses a k-vector
of p.g.f.s from a class of k-vectors of p.g.f.s, independently of the population, past
and present, and the previously selected k-vectors, which is then assigned to the
present population. Each particle of the present population then splits or disintegrates, independently of the others, in accordance with the p.g.f. assigned to its
classification. We call this process a multitype branching process in a random
environment (MBPRE).
In this work, we give some necessary and some sufficient conditions for almost
:
certain extinction of the MBPRE when there are at least two particle classifications.
I
To obtain these.results, we use Jensen's inequality, the dual process suggested by
Smith and Wilkinson (Ann. Math. Statist. 40, 814-827), and some results on products of
random matrices. Since our theorems require the user to evaluate limits of sequences
of products of random arrays, we include some corollaries which involve simpler,
albeit less general, conditions
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KEY WORD'
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Multitype Branching Processes
Generating Functions
Products of Random Matrices
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