* The work described in this report was done under Contract
N00014-67-A-032l-0002 with the Office of Naval Research
Branching Processes in Markovian Environments
by
Walter L. Smith *
University of North Carolina
Chapel Hill, N.C.
and
William E. Wilkinson
Duke University
Durham, N.C.
Institute of Statistics Mimeo Series No. 657
Department of Statistics, University of North Carolina at Chapel Hill
January 1970
Branching Processes in Markovian Environments
by
Walter L. Smith *
University of North Carolina
Chapel Hill, N.C.
Summary.
n
William E. Wilkinson
Duke University
Durham, N.C.
P
A positive recurrent Markov transition matrix
Markovian sequence
{Z}
and
{V }
n
of states called omens.
generates a
A branching process
develops in the usual way except that families born to the n-th
generation are all (independently) governed by a pgf
(s),
<Pi;;
o
~
s
~
1,
n
where
{z;; }
n
is the sequence of environmental variables.
depends on the omen
bution of
are otherwise independent.
V
The distri-
but the environment variables
n
A necessary and sufficient condition for
immortality of the process, that is
lim
n-+oo
P{Z
p
This assumes a particularly simple form when
only through its stationary distribution
n
{w.},
J
>
O}
>
0,
is given.
is finite, involving
say.
This simple form
of the condition is also shown to be sufficient for immortality, even
when
p
is infinite.
However, an example is given of two related pro-
cesses, which demonstrates that when
P
is infinite there cannot be a
necessary and sufficient condition for immortality involving no more
specific information about
tribution
*
p
than is contained in the stationary dis-
{w.}.
J
The work described in this report was done under Contract
NOOOl4-67-A-032l-0002 with the Office of Naval Research.
p
2
1.
Introduction.
In this paper, we shall generalize the theory given in Smith and
Wilkinson (1969) for the Branching Process with Random Environments
(BPRE) to cover situations where a certain kind of Markov dependence
holds between successive environmental variables.
We shall call the new
process a Branching Process with Markovian Environments (BPME).
P = lip lJ
.. 11
We imagine first an irreducible Markov transition matrix
= 1,2,3, ... )
(i,j
of a positive recurrent chain
{V},
n
~
which we shall
call the guiding chain; the states of this chain shall be called the
omens.
We shall assume
For each
of
iid
i
=
Va
1,2'00"
to be given and non-random.
let
{y(i,j)}
(j
=
1,2'00)
random variables taking values in some subset
t
dimensional Euclidean space , and call the points
environments.
be a sequence
of a finite
8
e in
8
the
Let us further assume that the random variables com-
prising the double sequence
{y(i,j)}
(i,j
=
1,2, ... )
are mutually
independent.
To simplify notation
as much as possible we shall write
for a random variable with the same d.f. as anyone of the
(j
=
1,2, ... ),
whenever no confusion can thereby arise.
represents a "typical" member of
y(i)
{y(i,j)}
Thus
y(i)
(1;; }
by the
{y(i,j)}.
We now define a sequence of environmental variables
n
equation
t
Since the set of all pgf's can be put into (1,1) correspondence
with a subset of the interval [0,1], the reader will soon see that 8
could be taken as uni-dimensional; we adopt, however, what seems a more
convenient assumption.
3
y(V , n+l),
(1.1)
0,1,2, . . • .
n
n
A very special case of the above process which merits particular
mention is the pure case.
This arises when the omen
termines the environment.
In other words, to each omen
e.1
corresponds an environment
all
= 1,2, ...
j
in
and
8
V
P{y(i,j)
uniquely de-
n
i
=
(i
e.}
1,2, .. )
1
1
for
If the BPME is not pure we shall, when the emphasis
is desirable, call it mixed.
Corresponding to each environment
o~
s
~
1,
e
E
0
is to be a pgf
of a non-negative integer-valued random variable.
¢e(s),
We can
then define the expected value of this random variable by
~
lim
(e)
s t 1
~(e)
and we shall assume that
We shall assume that
urable function of
(i)
for fixed
s,
e.
¢e(s),
{¢ y C1,].)(s)}
(ii)
is a sequence of iid variables;
=
1,2, •.•
00
for all
e
E
for each fixed
t
IV
s,
is a Borel meas-
Some consequences of this assumption are:
dependent random variables;
j
<
is a double sequence of mutually infor fixed
(iii)
s
and
{~(y(i,j))},
i,
i
{¢ ("
Y 1,J.)(s)}
1,2, ... ,
are mutually independent random variables and any subse-
quence obtained by holding
i
fast will be
a sequence of iid random
variables.
t
This for convenience; by suitable incantations about events of
probability zero it could be "generalized" to an a.s. condition.
4
We imagine the BPME to develop as follows.
2
0
particles and the omen for the environment is
Initially there are
V '
O
Every particle
in this zero-th generation then experiences an environment
So =
y(V ,l),
O
and the sizes of their families, though independent, are
to be governed by the same pgf
¢s (s).
o
In this way,
2
1
forming generation one, are created; the families of these
new particles,
2
1
particles
will then, in their turn, be produced under the influence of omen
VI'
The process then develops in an obvious way.
If
P{2
n
> O} ~ 0,
it is immortal.
we say the process
{2}
n
We shall see that mortality cannot be affected by
choice of the initial omen or by the initial size
be finite).
is mortal; otherwise
2
0
(so long as it
5
2.
Necessary and Sufficient conditions for Mortality of the BPME.
Let us select an arbitrary omen
i
the term i-cycZe we mean any finite sequence of omens
for which
vI
=
and
i
v
i
k :f
Va
and let us suppose
By
{v ,v 2 '··· ,vf}
l
k,
2
~
k
{,/i) }
r
'
r
=
1,2, ... ,
for every
i.
=
~
f.
We call
f
the Zength of the i-cycle.
Holding the omen
i
fixed, let
meration of the i-cycles.
be any enu-
Suppose that, in fact,
and define
(i)
Pr
p.
lV
p
2
p
v v
2 3
V
..
f
l
Since the guiding chain is positive recurrent, the omen
I t is clear that
almost surely, recur infinitely often.
i
(i)
Pr
will,
is the
probability that, given that the guiding chain is initially in omen
it will move through the omens of the i-cycle
omen
i.
Evidently
r
{Z
n
}
initial occurrence the zero-tho
~
i
we shall sayan epoch
... ,
occur at generations
= n(k+l)
~
- nk ·
in returning to
process with the aid of these i-cycles.
When the guiding chain enters omen
currence of
(i)
r
(i) = l.
r=l Pr
We shall study the
Let
If
i,
etc. ,
~
occurs.
and call its
Then between the k-th and (k+l)-th oc-
will be some i-cycle,
c
k
say, the length of which is
Let the environments encountered in this i-cycle be
... ,
6
and denote this vector by
kk.
Then we define
<jJ(i)(S)
~k
Routine, but tedious, measure-theoretic discussion will establish
that
{<jJ~i)(s)},
k
=
0,1,2, ... ,
is a sequence of independent and iden-
~k
tically distributed random variables for each fixed
s,
~
0
~
s
1.
If the guiding chain had only one state, the environmental variabIes
would merely be a sequence of iid random variables and the
{s}
~n
BPME would become the BPRE discussed in Smith and Wilkinson (1969).
The
latter paper coupled with Smith (1968) provide the following Theorem A.
Let a BPRE be based on iid environmental variables
in some abstract space
8,
with associated pgf's
be random variables for each fixed
\'00
Lr=O Pr(n)s
\'00
L r=
n
0 rp (s ).
r n
s,
be the Taylor expansion of
O~s~l.
¢s
n
(s)
taking values
{s },
n
{¢s (s)}
assumed to
n
Let
¢
sn
and set
E;,
(s)
n
=
To avoid triviality, assume the following conditions
We can now state:
THEOREM A,
Suppose that
p{ I;;
n
< oo}
=
1
and that
Elto91;;
n
I
< 00.
subject to A(i), (ii) above, it is necessary and sufficient for immortality of the BPRE that both the following conditions hold:
7
tog
~n > 0
C(i)
E
C(ii)
E tog [1 -
¢~
> _
(0)]
00
n
Let us now return to our BPME
{Z}.
It should be clear that, if
n
we observe this BPME only at successive epochs
tively observing a BPRE; i.e.,
Theorem A may be applied.
this BPRE is
{fk}
{Zn },
O~,
then we are effec-
k = 0,1,2, .•. ,
k
is a BPRE, and
The sequence of environmental variables for
and the corresponding sequence of pgf's is
from which the other relevant quantities must be derived.
For ease of writing, we shall let
pose
~(i)(s) =
a random variable.
that if
c-
<,
P sr.
\00
Lr=O
~(i)(S)
Necessarily, each
r
~(i)(S)
stand for
,.~O
and sup-
= 0,1,2, ... ,
r
is
It also transprires, after a fairly obvious argument,
\00
= Lr=l
rPr'
p{~ < oo}
then necessarily
= 1.
if we can
We are thus in a position to apply Theorem A to
find a suitable parallel to the triviality-avoiding conditions A(i),
(ii).
To this end, it is helpful at this point to introduce the station-
ary distribution
say,
{w },
v
v =
tive recurrent guiding chain.
has the property that
w
>
v
over the omens of the posi-
It is well known that this distribution
a
for all
event defined by a random omen
y(V).
1,2, ••• ,
v.
Suppose
A(V,y(V»
is any
V and the corresponding random variable
Then we shall write
00
P{A(V,y(V»}
w
v
P{A(v ,y(v»}.
v=l
Thus, if
Va
were assigned the stationary distribution
of being fixed at
i,
ability of the event
then P{A(V,y(V»}
A(V ,y(V ».
n
Borel function of the omen
n
V
and
v
instead
would be the stationary prob-
Similarly, i f
y(V),
{w }
f(V,y(V»
then we write
is any
8
00
Ef(V,y(V»
w Ef(v,y(v»
v
v=O
for the stationary expectation of
f(V ,y(V ».
n
n
We can now introduce
the following conditions on the BPME:
B(i)
Because
w
>
v
0
l{po(y(V»
for all
<
v, it
I}
= 1
will be seen that the above conditions
prevent the same trivial situations for the BPME as did A(i), (ii) for
the BPRE.
In particular, if the BPME satisfies B(i), (ii), it is easy
to see that the BPRE
2.1.
THEOREM
{Znk}
satisfies A(i), (ii).
Suppose that B(i), (ii) hold and that
We now prove:
Ellog~(y(V»1
<
00
Then it is necessary and sufficient for immortality of the BPME that the
following conditions both hold
D(i)
Elog
D(ii)
For some initial omen
~(y(V»
> 0
E log [1 - ~(i)(O)]
where
>
i,
_
00,
~(i)(s) is defined with reference to i-cycles as explained above.
PROOF.
the BPRE
It is apparent that the BPME
{Znk}
is immortal.
{Z }
n
is immortal if and only if
Thus we can deduce from Theorem A neces-
sary and sufficient conditions for immortality of the BPME.
C(ii) translates directly into D(ii).
translate as claimed, we prove:
Condition
To see that the other conditions
9
lEMMA 2,~
If we define the random variable
l_<jl(i) (s)
lim
1 - s
s t 1
and if we assume that
and:
Eliog~(y(V»
I
<
00
then:
(a) EI~og~1
<
00
(b) E ~og ~ = l/w. ££.og ~(y(V».
1
~,
Let us write
Then, since the guiding chain is
positive recurrent (though not neces-
sari1y aperiodic) there exists
n
1
lim
n
where the
{w.}
J
-+
00
w. ,
J
n
r=l
constitute the stationary distribution over the omens.
Of course, when the chain is aperiodic we can replace these Cesaro
limits by ordinary ones.
It is known that
n~i)(r)
cycle
TI
J
(i)
r
•
J
J
where
L.
is
J
Q..
the mean recurrence time of the epoch
Let
w. = l/L.,
J
be the number of occurrences of omen
j
in the i-
Then it can be shown that
00
p (i) n.(i) ( r )
(2.1)
r
J
r=l
a relation we shall need presently.
Suppose the initial i-cycle is
suppose that
~O
= (~O'~l""'~~-l)'
Co = (i,v1,v2, ... ,v~_1)'
Then, conditional upon
say,
Co
the
and
10
i:;. 's,
J
0 ::; j ::; t-l,
are independent random variables and
tribution depending only on the omen
~j = lims-+ l[l-¢i:;.(s)]/[l-s].
J
(with
v.
J
Then, given
of iterated pgf's we have that
-
V
1:;.
J
o = i) .
has a dis-
Let
by a familiar property
~o'
and that
~1~2" ·~t-l
t-l
tog
I
-
tog
~j
j=l
t-l
Itog:: I
Itog~
I
::;
.1
J
.
j=l
Thus, if
Co
TI
(i)
r
we see that
00
n~i)(r)Etog~(y(j»
(2.2)
J
j=l
00
I
(2.3)
j=l
n~i) (r)E!togUy(j»
J
1
.
From (2.3) we infer that
00
00
j=l
r=l
~
eltog~(y(V» I
w.
1.
Since
w. > 0,
1.
it is clear that (a) is proved.
To prove (b), we merely
repeat on (2.2) the argument we have used on (2.3) and appeal to the absolute convergence we have now proved to justify the reversal of the
double summation.
11
We now make some observations on Theorem 2.1.
Observation 1.
The theorem makes no reference to
ZO'
the initial
number of particles, which is merely supposed to be a (finite) strictly
positive integer.
This is reasonable because, from the theory of the
BPRE, we know that the BPRE is immortal if and only if it is so for the
special case
Thus, since the BPME is immortal if and only if
1.
Zo
the BPRE
{Zn } is immortal, then the BPME is immortal if
k
it is so when Z = 1.
0
Observation 2.
i.
v
Suppose
v
~d
only if
is any omen distinct from the initial omen
Since the guiding chain is irreducible positive recurrent, the omen
will almost surely arise after a finite number of generations.
Thus,
bearing Observation 1 in mind, if the BPME is mortal when it starts in
v
(with any initial number of particles) it must be mortal when its
initial omen is
i.
CoROLLARY 2,1,~
We, therefore, have the following:
If D(ii) holds for one initial omen
i
it holds for
any initial omen.
Observation 3.
The assumption
Ellog~(y(V»
I
<
00
a number of arguments in the theory of the BPRE.
is needed to justify
Suppose however that
and
while D(ii) is satisfied.
Select a large integer
6
and construct a
12
new "truncated" BPME from the given one as follows.
If
00
n=O
is a pgf of the given BPME then replace i t by the pgf
6-1
1/J
1;;
(s)
I
p (1;;)s
n
n
n=O
+
U"p
In other words, whenever a family size exceeds
duced by fiat to 6.
If
~'(y(V»
n
(+"
6
then imagine it re-
denotes a typical mean family size in
the "truncated" process then we can choose
Eltog~'(y(v»1 <
6
finite, but so large that
00
and
E tog
~'(y(v»
>
0 •
Clearly condition D(ii) is unaffected, so the "truncated" process will
be immortal.
Plainly, whatever the initial omen and
ZO'
the probabil-
ity of ultimate extinction of the "truncated" process will not be less
than that of the original process.
Thus the original process is immortal,
and we have:
COROLLARY
21~1
Condition D(i) can be replaced by
Theorem 2.1 and the two Corollaries 2.1.1 and 2.1.2 are the most
we have been able to achieve in the way of necessary and sufficient conditions for immortality of the BPME.
The obvious drawback to any
13
application of these results is the checking of D(ii); presumably such
a check would be difficult.
Until the counterexample described in the
next section was obtained, we devoted some considerable effort to a
search for a substitute for D(ii), one which would be easier to verify.
We are now inclined to believe that such a more amenable condition does
However, if we ask for reasonably verifiable sufficient con-
not exist.
ditions for immortality, the picture is brighter and we have the
following.
THEOREM 2.2,
Suppose the BPME satisfies B(i) and (ii) and that
ellog~(y(V)) I < 00.
a)
b)
If
e log
as
n +
Then
~(y(V)) ~ 0
the process is mortal, i.e.,
p{Z
O}
>
n
+
a
00;
If the following two conditions hold:
e log
(2.4)
~(y(V)) > 0,
(2.5)
then
p{Z
>
n
O}
tends to some strictly positive limit as
which will depend on
Note.
Zo
and
VA'
n
+
00,
i.e., the BPME is immortal.
As explained in Observation 3, we can again deal with the situ-
f 10g+~(y(V)) = +
ation where
Proof.
00,
provided
£llog-~(y(V))1
00
Part (a) of the theorem follows at once from Theorem 2.1.
prove part (b) we have to show that (2.5) implies D(ii).
for each
<
e
E
8
define
To
To this end,
14
Evidently~
(2.6)
Furthermore ~
¢e (0) + [l-¢e (O)]¢e (0)
112
1 - [l-¢e (O)][l-¢e (0)]
1
2
by an induction
and~
argument~
k
(2. 7)
1 -
Tr
[l-¢e (0)]
j
for
=
k
l~2~
..•
Construct a pgf
,¥~i)(s)
from the pgf's
{1jJe(s)}
in the same way
~O
as
<p
(i)(s)
is constructed from the pgf's {¢e(s)}
'sO
'¥(s)
as <I>(:i)~) stands for
stand for
follows that
so that
Hence~
if
Co
again denote the initial
i-cycle~
00
~
I
r=l
p~i)
E{ Ilog (l-'¥(O))
II Co
and~
for
ease~
From (2.6)
let
it
15
£(i)_l
00
r
L
E{ I
r=l
£09[1-1'
j=O
by (2.7), when we write
Sj
(0)]
I Co
for the length of the i-cycle
IT
(i)
r
But
the last expression must equal
00
00
I
~
nji)(r) EI£og[l-1'y(j)(O)]\
Wi
&1£og[l-1' ( )(0)]
Y v
I,
j=l
r=l
by (2.1).
Thus (2.5) implies D(ii) and the theorem is proved.
Observation 4.
It should be remarked that D(ii) does not imply (2.5)
and, in general, (2.5) is not necessary.
underscores this remark.
The counterexample of §3
However, when the guiding chain has a finite
number of omens we can show (2.5) is necessary, and we obtain the following satisfactory corollary.
CoROLLARY 2.2.1.
If the BPME satisfies B(i), and (ii) and if there are
only finitely-many omens then (2.4) and (2.5) are necessary and sufficient for immortality of the BPME.
PROOF,
We have merely to show that, when there are finitely many omens,
D(ii) implies (2.5).
But it is easy to see that
<p (i)
(0)
1'y(i,l) (0)
and hence that
>
_
00
16
Corollary 2.1.1 shows that if D(ii) holds for one omen
for all values of
i.
i
it holds
Thus
>
_ '"
i, and, since there are finitely many such, (2.5) must hold.
for all
Thus the Corollary is proved.
Let
A
n
be the event
{Z
n
>
a}.
We close this section with the
proof of a theorem which shows that when the BPME is immortal
V
n
n
are asymptotically independent.
THEOREM
as
and
A
n
2.3.
-7- " ' ,
If the guiding chain is aperiodic and if
then, for every
n
PROOF.
w
v
lim
n-+<x>
n
-7-
Zk = O.
~
Then, if
N be the least integer
is fixed and large, and
n >
P{Z =0 V =v}
n ' n
L
P{N=j, V =v}
n
j=l
~
;::::
I
P{N. , V =v}
n
J
j=l
;::::
where
~'
c> 0
-7-
cW '
v
n
(2.8)
> O}
p{V =v}.
n
Let the improper random variable
such that
n
v,
p{Z >0, V =v}
where
P{z
~
~'
L
I
j=l
r=l
is a second large number.
P{N=j, V.=r, V =v}
J
n
P{N=j, V.=r, V =v}
n
'
J
But
P{N=j, V.=r}P{V =vJV.=r}
J
n
J
~,
k
17
and, by the familiar properties of an ergodic chain, this means that,
as
n
-+
00,
P{N=j, V.=r, V =v}
J
w P{N=j, V.=r}.
-+
n
v
J
It therefore follows from (2.8) that
lim inf P{Z =0 V =v}
n ' n
n -+
2
b'
-+
b'
L
L
j=l
r=l
w
v
00
If we let
b
on the right, and then
00
lim inf P{Z =0, V =v}
n -+
n
00
b
-+
J
00,
we obtain
w P{N
2
n
P{N=j, V.=r}
< oo}
v
w(l-c).
v
Hence
(2.9)
lim sup p{Z =0, V =v}
n
p{Z
However, since
-+
n
>
n
n
~
w c.
v
00
O} +,
from the equation
00
L
p{Z >O}
n
p{Z >0, V =k}
n
n
k=l
we may infer that for any large fixed
c
b,
p{Z >0, V =k} + p{V >b}.
n
n
n
::;
k=l
Thus, from (2.9), which is true for all
v,
for any fixed
b
c
~
L
k=l
k+l
I::; b:
00
cW
k
+ lim inf P{Z n >0, Vn =l} +
n -+
00
I
k=b+l
wk '
18
and this implies, since
1,
W
\00
Lk=l
k
that
00
lim inf P{Z >0, V
n -+
Now let
(2.10)
6
-+
00
00
n
n
=t}
~
L
CWt + (l-c)
wk·
k=6+l
and we infer that, for any omen
lim inf P{Z >0, V
n -+
00
n
n
=t}
t,
CWt·
The theorem follows from (2.9) and (2.10).
A similar theorem will hold in the periodic case; its proof will
differ little from the above.
19
3.
Some important counterexamples.
The conditions for immortality of the BPME in Theorem 2.1 do not in-
volve the transition probabilities
(Pij)'
but only the numbers
{w. } ,
J
which, when the guiding chain is aperiodic, constitute the unique stationary distribution of the guiding chain.
Bearing this observation in
mind, and comparing Theorem 2.1 with Theorem A, one is tempted to conjecture that there are necessary and sufficient conditions for immortality of the
BPME
which involve the transition matrix
through the numbers
only
In this section, we shall show that such a
{w. }.
J
conjecture is false.
I 1Pij II
We shall construct two
BPME's
in all respects except for their guiding chains.
which are identical
In each case, the
guiding chain is aperiodic and, what is important, they have identical
stationary distributions
{w.}.
J
However, one of these BPME's is mortal,
the other immortal.
Therefore, we are forced to conclude that any necessary and sufficient condition for immortality of the BPME must involve more specific
information about the matrix
arydistribution
{w }
j
I Ipijl I
than is contained in the station-
alone.
We find the existence of these two BPME's somewhat surprising.
The long-term averages of occurrence of the various
pgf's
of family
size are the same in each process, but in some strange way it is the
pattern of these occurrences that is crucial.
For convenient reference, we shall refer to these two processes as
the ladder process and the mock process, the choice of names having been
suggested by their particular constructions.
20
Ladder Process.
Let
{V}
be a Markov chain on the state space
m
consisting of the positive integers and with a transition probability
matrix of the form
l-r
e
l-r
l-r
The probabilities
{r. }
l
1
2
3
r
0
r
0
0
0
2
r
(3.1)
where
c
\'00
3
= 1/(Ln=1
lin ).
3
are defined by the equations
1 - r
r r
l 2
.. l
0
0
l
...
c
1
r 1 (l-r 2 )
c/2
r r (l-r 3)
l 2
c/3
r n- 1 (l-rn )
cln
Adding the first
3
3
3
equations in (3.1), we
n
obtain
n
L
c
Iii
3
,
i=l
so that
00
(3.2)
l/i
c
3
.
i=n+l
Since the right-hand side of (3.2) is strictly positive for all
follows that
r. > 0
l
for all
i.
n,
it
From the equations (3.1), it follows
21
that
P
l-r.
a
>
1
for all
i.
a
Hence
<
r.
<
l.
1
for all
so that
is indeed a transition probability matrix.
We now want to define a pure
We shall assume that
Va =
1
BPME
with
{V}
I-cycle
{1,2, ... ,n-l},
and, for notational convenience, shall
If
3
c/n ,
determined by omen
<P (s)
n
TI
represents the
n
then
(3.3)
pgf
as the guiding chain.
m
drop the superscript denoting the initial omen.
The
i,
n
= 1,2, ...
n
is defined by
if
n
=
if
n
= 2,3 ..•
1
<P (s)
n
where, with
A
n
{x}
= {2e n+ l £og2}.
so that
denoting the greatest integer in
Thus
¢'(l) > 1
<Pi (1) =
for
n
~
n
¢'(l)
n
Furthermore, if
2
~
n
¢'(l) = e
and
-n
n
{2e
n+l
and
2elog2
n -+
as
00
is defined by
¢ (s)
n
¢ (s)
1
x+l,
=
we have
n
¢'
n
(1)
IT
.
·+1
e- J {2e J
wg2} ,
j=2
from which it follows that
log¢'(l)
n
~
(n-l)log(2elog2),
as
n
-+
00.
log2},
n
~
2,
22
Hence, using (3.3), we see that
P
n
The series
log¢'
3
n
'\00
Ln=l
(1) ~ c(n-l)log(2elog2)/n ,
plog¢'(l)
n
n
as
n
->-
00
therefore converges; its sum is clearly
positive.
If for each positive integer
n,
we define
then
log y
Hence for all
n
~
n,
y
n
<~.
n
J.r:)...
ne
-(n+l)
<
-
2
<
-
log 2.
Using this inequality, we see that for
2,
<
Since
¢ (0)
n
< l-~e
-n
<
<
,
1 - ~e
we obtain
-n
•
23
<
<
¢2 (l-~e
<
1 -
~e
-3
)
-2
Hence
¢ (0)
¢1(¢2(···¢n(0) ... »
n
¢2(¢3(···¢n(0) ... »
<
1 - ~e
-2
,
so that
fog(l-¢ (0»
n
> fog~e
-2
,
and, consequently,
00
I
p fog(l-¢ (0»
n
n
> fog~e
-2
>
_00
n=l
It therefore follows from Theorem A that the ladder process is immortal.
If
{v },
m
{w}
n
is the stationary distribution of the guiding chain
then
00
I
w
n
00
I
Pi l
i=n
ip.
1
i=l
00
6
TI 2
I
I/i
3 ,
i=n
since
c 11·3
and thus
oo
l.
1=
1 ip.1 =
2
TI
c/6.
We note here, for later
24
reference, that
w
l
> ~.
Furthermore,
00
6
w
n
TI
2
x
f
-3
3
2 2
TI n
dx
n
so that
3
- -2TI n
w log(l-¢ (0»
n
n
and therefore
00
I
_
w tog (l-¢ (0»
n
00
n
n=l
Mock Process.
For this process, the guiding chain
{v }
m
is de-
fined as a Markov chain on the state space consisting of the positive
integers and with a transition probability matrix of the form
a
p
=
o
1
n
o
100
We again assume that
represented by
Pn
TIl
= {l}
V = 1;
o
and
TI
n
the l-cycles in this case may be
- {l, n} ,
n = 2,3,... .
The probability
associated with the n-th l-cycle is then given simply by
n = 1,2, . . . .
If
n. (r)
J
is the number of occurrences of omen
j
in
then
26
In particular, then, we can choose the sequence
{a.}
so that the
J
guiding chain of the mock process has the same stationary distribution
as has the guiding chain of the ladder process.
{w. }
J
thereby obtained, we define a BPME with
{a. }
J
by using the same sequence
{¢ (s)}
n
{v }
m
With the sequence
as guiding chain
of pgf's as were defined in the
ladder process.
For the mock process, we have
<P (s)
since
Since
n
=
¢l(s) = s.
Thus
<P i (1)
i'n (1)
e
w
n
~
2 2
3/rr n ,
-n
{2e
n+1
10g2}
a
and thus
\'00
L n =l
Pnlogi'n (1)
log
n;::: 2,
and for
~
n ;::: 2,
2elog2,
as
n
-+
00
it follows that
(3.5)
for
1,
n
is positive and finite.
(1-<I> (0))
n
log (1-¢ (0))
Furthermore, since
- n
n
it follows from (3.5) that
00
I
Pn
log (l-<P n (O))
00
n=l
By Theorem A, the particular mock process we have considered is mortal.
27
REFERENCES
Smith, Walter L. (1968),
Necessary conditions for almost sure extinction
of a branching process with random environment,
39,
Ann. Math. Statist.
2136-2140.
Smith, Walter L. and Wilkinson, William E. (1969),
cesses in random environments,
On branching pro-
Ann. Math. Statist.
40,
814-827.
Unc.lassitieo
Security ClassificatIon
-y
14
LIN K
LIN K
A
LIN K
8
C
KEYWORDS
WT
ROLE
ROLE
WT
WT
ROLE
Branching Processes
Markov Processes
Probability Generating Functions
I
I
I
II
~
Security ClassificatIon
I
I
I
UNCT.ASSIFIED
Secunty CI aSSlT lcallon
DOCUMENT CONTROL DATA - R&D
(Security classification 01 title, body of abstract and ;ndex;n~ annotation must be entered when the overall report Is classified)
ORIGINATING ACTIVITV
I
(Corporate author)
,I
2_. REPORT SECURITY CLASSIFICATION
Department of Statistics
University of North Carolina
Chapel Hill, N.C. 27514
Unclassified
2b.
GROUP
REPORT TITLE
l
BRANCHING PROCESSES IN MARKOVIAN ENVIRONMENTS
4. DESCRIPTIVE NOTES (Type
of report and inclusive dates)
Technical Report
5·
AU THORISI
(First name, middle initial, last name)
Smith, Walter Laws
6.
and
Wilkinson, William E.
REPORT DATE
7a. TOTAL NO. OF PAGES
January 1970
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rb.
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c.
OTHER REPORT NOIS) (Any other
numbers that may be assigned
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Distribution of the document is unlimited
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SUPPLEMENTARY NOTES
12. SPONSORING MILITARY ACTIVITY
Logistics and Mathematical Statistics
Branch Office of Naval Research
Washington. D.C. 20360
\3
ABSTRACT
A positive recurrent Markov transition matrix
p
generates a Markovian sequence{V }
n
of states called omens. A branching process {Z } develops in the usual way except that
n
families born to the n-th generation are all (independently) governed by a pgf ¢Z;;n(s),
O~s~l,
where {Z;; } is the sequence of environmental variables. The distribution of r;;
n
n
depends on the omen V but the environment variables are otherwise independent. A necn
essary and sufficient condition for immortality of the process, that is lim oop{Z >O}
n-+
n
>0, is given. This assumes a particularly simple form when p is finite, involving
~
E
~
only through its stationary distribution {w.} , say.
J
This simple form of the condition
is also shown to be sufficient for immortality, even when
p is
infinite.
example is given of two related processes, which demonstrates that when
However, an
p is
infinite
I
,
I
i
there cannot be a necessary and sufficient condition for immortality involving no more
specific information about
p than
is contained in the stationary distribution {w.} •
J
~
,
Security Classification