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On the supercritical Galton-Watson process with immigration
MATHEMATICAL
9
BIOSCIENCES
On the Supercritical Immigration *
Galton-Watson
Process with
E. SENETAT StatisticaI Laboratory, University of Cambridge Cambridge, England Communicated
by Richard Bellman
ABSTRACT It is shown for a supercritical Galton-Watson process with immigration, {X,}, that there exists a sequence of constants {c,,} such that the process {XJc,,} converges almost surely to a random variable with continuous distribution concentrated on (0, co), or diverges almost surely to infinity according as the log moment z? b, log j of the immigration distribution is finite or not. These results include the suggested refinements proposed in a previous paper of the author [6].
1. INTRODUCTION
We confine ourselves to a very brief exposition on the process, directing the reader to the author’s previous paper on this topic and the references therein for greater detail. For n > 1, x, = z, + r-J;’ + . * . + u?), (1) where Z, is the number of direct descendants of the initial individual in the (&I = l), and Ut’, i = 1, . . . , n is the number of descendants 12th generation from immigration at the ith. Thus {Z,} is an ordinary Galton-Watson process, and the component variables of X,, in (1) are independent. The offspring and immigration probability distributions, defined on the nonnegative integers and denoted correspondingly by * Work supported by a Nuffield Dominion Travelling Fellowship. t Permanent address: Department of Statistics, Australian National University, Canberra. Mathematical Biosciences 7 (1970), 9-14 Copyright @ 1970 by American Elsevier Publishing Company,
Inc.
10
E. SENETA
{b,}, have “cumulant generating functions” (c.g.f.‘s) defined, respectively, by k(s) = -log E[exp -s&1, Y(S) = -log E[exp --sUF)], s > 0. They are assumed to satisfy f,# 1, j > 0, and b, < 1. We are concerned only with the situation 1 < ~1 3 2 jjj < co. In a recent paper [5] the author has shown that there exists a sequence of positive constants {c,} such that the sequence {2,/c,} converges in distribution to a proper nondegenerate random variable W. Heyde [2] has remarked that the distribution of W is continuous on (0, co) and, by is a martingale, has shown that noting that the sequence {exp( -2,/c,,)} The stated continuity follows essenconvergence of {Z,Jc,,} is almost sure. tially from an argument of Stigum [7] ; and Heyde’s brief argument regarding the martingale, which we shall need to imitate in the sequel, depends heavily on the nature of the constants (c,}, whose relevant properties we now note, from our earlier paper [5]. The constants c, are given by c, = l/h,(s,), where s0 is an arbitrary (but “small”) fixed positive number, and h,(s) is the inverse function of h,(s) and k,(s) are the k,(s) E --log E[exp -sZ,], s > 0. Moreover nth functional iterates of h,(s) and k,(s), respectively. It follows that -+ m as n -+ 00, and that, for n 2 no(e), c?l + 009 c,&, = Z&)l~,+,(~,) {fj},
1 n
0
Cl ;
Q h,(G)
<
11 + -
c2
m
c
El n
(2)
3 1
where E, c1 and c2 are positive constants and 1 + E < m, as may be readily seen from the mean-value theorem, so that the constants {cn} are geometrically bounded. Let us now write
w,=G,
z =
(lp
+ UF) + * . . + C?I
C,
Then,
uy>
n
in terms of c.g.f.‘s, for s > 0, t,,+,(s) z -log
E[exp --sZ,+,]
=
an equation of which we shall make repeated use, and which follows directly from the definition of Z,, or from the evolutionary equations of {X,} given in several of the previous papers (see, e.g., [6]). We shall prove, in the sequel, several extensions of the results of our previous paper [6] precisely in the general directions indicated in Section 3 of that paper, in particular using the norming sequence {c,}. The whole development rests heavily on (i) noting that the process (-exp(-xX,/c,)} is a submartingale; and (ii) using the basic techniques of the earlier treatment. Mathematical
Biosciences
7 (1970), 9-14
GALTON-WATSON
I1
PROCESS
In relation to this earlier paper also, we wish to note that its main result, via a reformulation, is deducible from Theorem 2.1 of Kesten and Stigum [3] on an ordinary decomposable multitype Galton-Watson process. This point will be taken up and generalized elsewhere; however, as already noted, the method of the earlier paper makes the treatment of the sequel more economical.
2. PRINCIPAL
RESULTS
Let us write V, = X,/en = h,(s,,)X,. -log
Then
v,+11I v,> = -log E(exp [-~,+,(~oK+I1 1x,> = ~(4l+1(%)) + ~?&@,+,w > uL(%) = Vn,
E(exp [-
whence the sequence {-exp [- V,]} is a submartingale, since {V,} is Markovian. Moreover, since V,, > 0, the conditions of the submartingale convergence theorem (Lotve [4, p. 3931) are satisfied for this submartingale sequence. Hence the sequence of interest, {V,,}, with probability 1 approaches a random variable V satisfying 0 Q V Q co. Let us put
P = Pr[V < cc]. We next showp
= 1 or 0. Since, as noted in Section I,=kI=
v-
1, W,, &.8. W( < co),
w,
where Pr[Z < co] = p. Moreover,
from (3),
tn+~(s) = L(z) whence,
+ r(k,($-))
for s > 0, letting n --+ co, T(s)
=
T(t)
+
r(K(i)),
(4)
where T(s) = -log
E[exp --sl],
K(s) = --log E[exp -SW],
where the first of these two expectations only. Iterating (4) yields
is taken
over finite values
of Z
T(s)= T(S)+Y@(>)), Mathematical
Biosciences
7 (1970), 9-14
E. SENETA
12 whence
j_;((&I)). K
T(s) = no+) + i On the other hand,
(5)
from (3) also,
tn+l(s)= i
(6)
r(kn-j(shn+l(sO))),
j=O
whence
whence
Now, either Z”(1) < co, in which case T(s) < co for all s > 0; or T(1) = co, in which case T(s) - co for all s > 0 and p = 0. The former case occurs if and only if ~~Chj+l(sO))
which,
on account
of the geometric ib,logj
<
(7)
*9
bounds
(2), is equivalent
to
< co
j=l
along the lines of the lemma of Heathcote, Seneta and Vere-Jones [ll. Moreover, if (7) holds, then, for the value s = 1 in (6), letting IZ- ~IJ in this equation, by dominated convergence,
whence, from (5), T(O+) = 0 making I < co almost everywhere, and p = 1. Thus, to prove our main assertion, we need only verify that, in the case p = 1, V has a continuous distribution concentrated on (0, 00). This follows from the properties of Wand I as in [6], using in particular a result of Smith. Additional Properties. Using the methods and results of Seneta [5, 61, it is easy to relate the norming {cn} to the previously used norming {HP}, and also to discuss to some extent such properties of the random Mathematical Biosciences
7 (1970). 9-14
GALTON-WATSON
13
PROCESS
variables V, W, and I as the finiteness of expected values; for example (in the case p = l), that E[V] < CCG= E[Z, 1ogZJ < co and il < co, where )3 s E[Ue’] = 2 jbi, is an easy deduction. A rather more subtle question requiring an answer, in view of the preceding development, is whether it is always possible to find a positive norming sequence, {E,}, say, such that (X,/C,} always converges (at least in distribution) to a proper nondegenerate random variable, irrespective of conditions on the immigration distribution {bj}. We now show that no such sequence satisfying the condition c?L+1 -+m Cl
as
n-+co
may exist. For suppose it does: then along development, for s ) 0, with the corresponding r-l i,+&) (
-
i n(z)\
where r-l{.} is the inverse function r-l(T(s)
-
T(i))
the lines of our previous modification of notation
= kn(GJ9
of r{.}.
Letting
n -+ CO,
= R(~)Eli~k,(&).
Moreover, by continuity at the origin, R(s) is the c.g.f. of a proper random variable, which moreover cannot be degenerate, since the left-hand side of the above expression is not identically zero or linear in s. (To see the last point proceed as follows. If K(s) = 0, then
T(s) -
T
02 = m
0,
which results in a contradiction to the assumption (c > 0), then, as before in obtaining (5), T(s)
=$r j=o
about T(s). If R(s) = C.S.
(--$ >,
which diverges for all s > 0 if 2: bj log j = co, which is again a contradiction to our assumptions about T(s).) By a result of Seneta [5], it must therefore follow that asn+co, En - const c, which by the preceding results of the present paper yields a contradiction to the assumed nature of {E,}. Mathematical
Biosciences
7 (1970), 9-14
14
E. SENETA
REFERENCES 1 C. R. Heathcote, E. Seneta and D. Vere-Jones. A refinement of two theorems in the theory of branching processes. Tar. Veroyuf. Primenen, 12(1967), 341-346. 2 C. C. Heyde, private communication, 1969. 3 H. Kesten and B. P. St&urn, Limit theorems for decomposable multi-dimensional Galton-Watson processes, J. Math. Anal. Applications, 17(1967), 309-338. 4 M. Loeve, Probubilify Theory, Van Nostrand, Princeton, N.J., 1963. 5 E. Seneta, On recent theorems concerning the supercritical Galton-Watson process, Ann. Math. Statist., 39(1968), 2098-2102. 6 E. Seneta, A note on the supercritical Galton-Watson process with immigration. Math. Biosci. 6(1970), 305-312. 7 B. P. Stigum, A limit theorem on the Galton-Watson process, Ann. M&h. Statist., 37(1966), 695-698.
Mathematical
Biosciences
7 (1970), 9-14
9
BIOSCIENCES
On the Supercritical Immigration *
Galton-Watson
Process with
E. SENETAT StatisticaI Laboratory, University of Cambridge Cambridge, England Communicated
by Richard Bellman
ABSTRACT It is shown for a supercritical Galton-Watson process with immigration, {X,}, that there exists a sequence of constants {c,,} such that the process {XJc,,} converges almost surely to a random variable with continuous distribution concentrated on (0, co), or diverges almost surely to infinity according as the log moment z? b, log j of the immigration distribution is finite or not. These results include the suggested refinements proposed in a previous paper of the author [6].
1. INTRODUCTION
We confine ourselves to a very brief exposition on the process, directing the reader to the author’s previous paper on this topic and the references therein for greater detail. For n > 1, x, = z, + r-J;’ + . * . + u?), (1) where Z, is the number of direct descendants of the initial individual in the (&I = l), and Ut’, i = 1, . . . , n is the number of descendants 12th generation from immigration at the ith. Thus {Z,} is an ordinary Galton-Watson process, and the component variables of X,, in (1) are independent. The offspring and immigration probability distributions, defined on the nonnegative integers and denoted correspondingly by * Work supported by a Nuffield Dominion Travelling Fellowship. t Permanent address: Department of Statistics, Australian National University, Canberra. Mathematical Biosciences 7 (1970), 9-14 Copyright @ 1970 by American Elsevier Publishing Company,
Inc.
10
E. SENETA
{b,}, have “cumulant generating functions” (c.g.f.‘s) defined, respectively, by k(s) = -log E[exp -s&1, Y(S) = -log E[exp --sUF)], s > 0. They are assumed to satisfy f,# 1, j > 0, and b, < 1. We are concerned only with the situation 1 < ~1 3 2 jjj < co. In a recent paper [5] the author has shown that there exists a sequence of positive constants {c,} such that the sequence {2,/c,} converges in distribution to a proper nondegenerate random variable W. Heyde [2] has remarked that the distribution of W is continuous on (0, co) and, by is a martingale, has shown that noting that the sequence {exp( -2,/c,,)} The stated continuity follows essenconvergence of {Z,Jc,,} is almost sure. tially from an argument of Stigum [7] ; and Heyde’s brief argument regarding the martingale, which we shall need to imitate in the sequel, depends heavily on the nature of the constants (c,}, whose relevant properties we now note, from our earlier paper [5]. The constants c, are given by c, = l/h,(s,), where s0 is an arbitrary (but “small”) fixed positive number, and h,(s) is the inverse function of h,(s) and k,(s) are the k,(s) E --log E[exp -sZ,], s > 0. Moreover nth functional iterates of h,(s) and k,(s), respectively. It follows that -+ m as n -+ 00, and that, for n 2 no(e), c?l + 009 c,&, = Z&)l~,+,(~,) {fj},
1 n
0
Cl ;
Q h,(G)
<
11 + -
c2
m
c
El n
(2)
3 1
where E, c1 and c2 are positive constants and 1 + E < m, as may be readily seen from the mean-value theorem, so that the constants {cn} are geometrically bounded. Let us now write
w,=G,
z =
(lp
+ UF) + * . . + C?I
C,
Then,
uy>
n
in terms of c.g.f.‘s, for s > 0, t,,+,(s) z -log
E[exp --sZ,+,]
=
an equation of which we shall make repeated use, and which follows directly from the definition of Z,, or from the evolutionary equations of {X,} given in several of the previous papers (see, e.g., [6]). We shall prove, in the sequel, several extensions of the results of our previous paper [6] precisely in the general directions indicated in Section 3 of that paper, in particular using the norming sequence {c,}. The whole development rests heavily on (i) noting that the process (-exp(-xX,/c,)} is a submartingale; and (ii) using the basic techniques of the earlier treatment. Mathematical
Biosciences
7 (1970), 9-14
GALTON-WATSON
I1
PROCESS
In relation to this earlier paper also, we wish to note that its main result, via a reformulation, is deducible from Theorem 2.1 of Kesten and Stigum [3] on an ordinary decomposable multitype Galton-Watson process. This point will be taken up and generalized elsewhere; however, as already noted, the method of the earlier paper makes the treatment of the sequel more economical.
2. PRINCIPAL
RESULTS
Let us write V, = X,/en = h,(s,,)X,. -log
Then
v,+11I v,> = -log E(exp [-~,+,(~oK+I1 1x,> = ~(4l+1(%)) + ~?&@,+,w > uL(%) = Vn,
E(exp [-
whence the sequence {-exp [- V,]} is a submartingale, since {V,} is Markovian. Moreover, since V,, > 0, the conditions of the submartingale convergence theorem (Lotve [4, p. 3931) are satisfied for this submartingale sequence. Hence the sequence of interest, {V,,}, with probability 1 approaches a random variable V satisfying 0 Q V Q co. Let us put
P = Pr[V < cc]. We next showp
= 1 or 0. Since, as noted in Section I,=kI=
v-
1, W,, &.8. W( < co),
w,
where Pr[Z < co] = p. Moreover,
from (3),
tn+~(s) = L(z) whence,
+ r(k,($-))
for s > 0, letting n --+ co, T(s)
=
T(t)
+
r(K(i)),
(4)
where T(s) = -log
E[exp --sl],
K(s) = --log E[exp -SW],
where the first of these two expectations only. Iterating (4) yields
is taken
over finite values
of Z
T(s)= T(S)+Y@(>)), Mathematical
Biosciences
7 (1970), 9-14
E. SENETA
12 whence
j_;((&I)). K
T(s) = no+) + i On the other hand,
(5)
from (3) also,
tn+l(s)= i
(6)
r(kn-j(shn+l(sO))),
j=O
whence
whence
Now, either Z”(1) < co, in which case T(s) < co for all s > 0; or T(1) = co, in which case T(s) - co for all s > 0 and p = 0. The former case occurs if and only if ~~Chj+l(sO))
which,
on account
of the geometric ib,logj
<
(7)
*9
bounds
(2), is equivalent
to
< co
j=l
along the lines of the lemma of Heathcote, Seneta and Vere-Jones [ll. Moreover, if (7) holds, then, for the value s = 1 in (6), letting IZ- ~IJ in this equation, by dominated convergence,
whence, from (5), T(O+) = 0 making I < co almost everywhere, and p = 1. Thus, to prove our main assertion, we need only verify that, in the case p = 1, V has a continuous distribution concentrated on (0, 00). This follows from the properties of Wand I as in [6], using in particular a result of Smith. Additional Properties. Using the methods and results of Seneta [5, 61, it is easy to relate the norming {cn} to the previously used norming {HP}, and also to discuss to some extent such properties of the random Mathematical Biosciences
7 (1970). 9-14
GALTON-WATSON
13
PROCESS
variables V, W, and I as the finiteness of expected values; for example (in the case p = l), that E[V] < CCG= E[Z, 1ogZJ < co and il < co, where )3 s E[Ue’] = 2 jbi, is an easy deduction. A rather more subtle question requiring an answer, in view of the preceding development, is whether it is always possible to find a positive norming sequence, {E,}, say, such that (X,/C,} always converges (at least in distribution) to a proper nondegenerate random variable, irrespective of conditions on the immigration distribution {bj}. We now show that no such sequence satisfying the condition c?L+1 -+m Cl
as
n-+co
may exist. For suppose it does: then along development, for s ) 0, with the corresponding r-l i,+&) (
-
i n(z)\
where r-l{.} is the inverse function r-l(T(s)
-
T(i))
the lines of our previous modification of notation
= kn(GJ9
of r{.}.
Letting
n -+ CO,
= R(~)Eli~k,(&).
Moreover, by continuity at the origin, R(s) is the c.g.f. of a proper random variable, which moreover cannot be degenerate, since the left-hand side of the above expression is not identically zero or linear in s. (To see the last point proceed as follows. If K(s) = 0, then
T(s) -
T
02 = m
0,
which results in a contradiction to the assumption (c > 0), then, as before in obtaining (5), T(s)
=$r j=o
about T(s). If R(s) = C.S.
(--$ >,
which diverges for all s > 0 if 2: bj log j = co, which is again a contradiction to our assumptions about T(s).) By a result of Seneta [5], it must therefore follow that asn+co, En - const c, which by the preceding results of the present paper yields a contradiction to the assumed nature of {E,}. Mathematical
Biosciences
7 (1970), 9-14
14
E. SENETA
REFERENCES 1 C. R. Heathcote, E. Seneta and D. Vere-Jones. A refinement of two theorems in the theory of branching processes. Tar. Veroyuf. Primenen, 12(1967), 341-346. 2 C. C. Heyde, private communication, 1969. 3 H. Kesten and B. P. St&urn, Limit theorems for decomposable multi-dimensional Galton-Watson processes, J. Math. Anal. Applications, 17(1967), 309-338. 4 M. Loeve, Probubilify Theory, Van Nostrand, Princeton, N.J., 1963. 5 E. Seneta, On recent theorems concerning the supercritical Galton-Watson process, Ann. Math. Statist., 39(1968), 2098-2102. 6 E. Seneta, A note on the supercritical Galton-Watson process with immigration. Math. Biosci. 6(1970), 305-312. 7 B. P. Stigum, A limit theorem on the Galton-Watson process, Ann. M&h. Statist., 37(1966), 695-698.
Mathematical
Biosciences
7 (1970), 9-14