CONTINUITY FOR MULTI-TYPE BRANCHING PROCESSES WITH ...

January 5, 1998

CONTINUITY FOR MULTI-TYPE BRANCHING PROCESSES WITH VARYING ENVIRONMENTS OWEN DAFYDD JONES, Australian National University

Abstract

Conditions are derived for the components of the normed limit of a multi-type branching process with varying environments, to be continuous on (0 1). The main tool is an inequality for the concentration function of sums of independent random variables, due originally to Petrov. Using this, we show that if there is a discontinuity present, then a particular linear combination of the population types must converge to a non-random constant (Equation (1)). Ensuring this can not happen provides the desired continuity conditions. BRANCHING PROCESS; MULTI-TYPE; VARYING ENVIRONMENT; CONTINUITY AMS 1991 SUBJECT CLASSIFICATION: PRIMARY 60J80 SECONDARY 60G17 ;

1. Introduction A multi-type branching process with varying environment (MTBPVE) generalises the classical multi-type branching or Galton-Watson process. For a nite number d of types, we allow the number of type j ospring of a type i parent at time n to depend on i, j and n. In Jones (1997), conditions were given for the L2 and a.s. convergence of an MTBPVE normed by its mean. It what follows, we derive natural conditions for the components of that limit to be continuous on (0; 1). Before proceeding, we need to introduce some notation. For a matrix A 2 Rdd , write A(i; j ) for its (i; j )th element, A(i;
) for the row vector given by its ith row and A(
; j ) for the column vector given by its j th column. Similarly, for a vector a 2 Rd write a(i) for its ith component. The vector of 1s will be written 1 and the unit vector with a 1 in position i will be written ei . Suppose that the ospring distributions of our branching process are given by a sequence of Zd+d valued r.v.s fXn g1 n=0 . That is, the distribution of the number of type j children born to a single type i parent at time n is the same as that of Xn (i; j ). De ne Mn = E Xn . We will assume that the Mn are nite in all that follows. For xed m  0, let Zm = fZm;n g1 n=m be the branching process de ned in the usual way (see for example Asmussen & Hering (1983) or Athreya & Ney (1972)), letting Zm;n (i; j ) be the number of type j descendants at time n of a single type i parent at Postal address: Centre for Mathematics and its Applications, Australian National University, ACT 0200, Australia. 

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2

Owen Dafydd Jones

time m. Note that as de ned, Zm takes on values in Zd+d, where the rows of Zm are independent processes. For a sequence of matrices fAn g1 n=0 , we will write Am;n for the forward product from m to n ? 1. That is, Am;n = Am Am+1


An?1 . It follows from the branching property of Zm that for any m  n  p, E(Zm;p jZm;n ) = Zm;n Mn;p: We will make the following assumptions about Zm . A There exist non-negative diagonal rescaling matrices f mRn g0mn , mRn = diag( mRn (1); : : : ; mRn (d)), such that for all m, a:s: L 1T as n ! 1; Zm;n mRn?1 ?! m

where Lm  0 and wm := E Lm is a strictly positive probability vector. B There exist non-zero, nite scalars fmn g0m;n such that for all m and n, mR nR?1 ! m I p p n

as p ! 1:

C For all i, mRn (i) ! 1 as n ! 1. Conditions sucient to ensure that A and B hold are given by Jones (1997) Theorems 1 and 2. Now, from A and B it follows that m ?1 n ?1 n m ?1 T n wm 1T = plim !1 Mm;p Rp = Mm;n plim !1 Mn;p Rp ( Rp Rp ) = Mm;n wn 1 m :

Thus, in the terminology of Cohn & Nerman (1990), fnm wn g1 n=m is a harmonic . So, if we let F be the - eld generated sequence for the matrices fMm;n g1 m;n n=m n by fZm;k gk=m , then E(nm+1 Zm;n+1 wn+1 jFm;n ) = nm Zm;n wn : 1 That is, fnm Zm;n wn g1 n=m is a martingale w.r.t. the ltration fFm;n gn=m . In particular, as E nm Zm;n wn = wm > 0, (nm wn (i))?1 must grow at least as fast as mRn (i). Thus, as wn is a probability vector, (n )?1 = m must grow at least as fast n m as the slowest of the mRn (i), as n ! 1.

2. Continuity De ne

un = 1max id

sup

l6=02R+;a2R+ d

P(Xn (i;
)l = a):

Continuity for branching processes

3

Proposition 2.1 (Continuity of the limit) Suppose that Conditions A, B and C hold, and that 0n (1 ? un ) ! 1. Then Lm (i) is continuous on (0; 1) for all i and m  0. Proof. The proof follows that of Cohn (1996) for the single-type varying environment case. See also Cohn & Jagers (1994). W.l.o.g., we will consider only L0 (1). Let be the set of sample paths of Z0 (1) := fZ0;n (1;
)g1 n=0 . For each n  0, let Fn be the - eld induced by Z0;n(1;
), let F1 be the smallest - eld containing all of the fFn g1 n=0 , and let P be the measure on ( ; F1 ) de ned by Z0 (1). Suppose that L0 (1) has a jump point at some c > 0. To begin with, we will show that there exist ln 2 Rn+ and an 2 R+ , n = 0; 1; : : : , such that (1) 1fL (1)=cg = nlim !1 1fZ (1;
)l =a g 1 a.s. 0

0;n

n

n

where

 = fL0 (1) 2 (0; 1)g: Let n = P(L0 (1) = cjZ0;n (1;
)), then fn g1 n=0 is a bounded martingale which thus converges a.s. and in mean to P(L0 (1) = cjF1 ) = 1fL (1)=cg . (This martingale construction is due to Doob. See for example Equation (9) x12.3 of Grimmett & Stirzaker (1992).) We have that for any n  0 0

(1;j ) d Z0X X n Ln(p) (j ) L0(1) = 0 j =1 p=1 ;n

a.s.

where the Ln(p) (j ) are i.i.d. as Ln (j ). Thus if ~z = fzn g1 n=0 2 is a realisation of Z0 (1) for which L0 (1)(~z ) = c, then xing ~z, 1 = nlim !1 n (~z) = nlim !1 P(L0 (1) = cjZ0;n (1;
) = zn ) n = nlim !1 P(0

(2)

(j ) d zX X n

j =1 p=1

Ln(p) (j ) = c):

Also, as c > 0, ~z 2  and so for all j , 0 1 fzn (j )g1 n=0  f Rn (j )gn=0 : (Here we write fan g  fbn g if lim an =bn exists 2 (0; 1).) Since 0Rn (j ) ! 1, this implies that zn (j ) ! 1 for all j . Now, it can be shown (using, for example, the method of Petrov (1975) Chapter III Theorems 3 and 4 applied to his Inequality (1.7)) that there exists an absolute constant A such that for any independent random variables U1 ; : : : ; Un (3)

sup P( x

n X i=1

Ui = x)  A

n X i=1

(1 ? sup P(Ui = x)) x

!?1=2

:

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Owen Dafydd Jones

Thus, for all n  0 (4)

2 32 2 d 3 (j ) d zX X X 4sup P( Ln(p) (j ) = x)5 4 zn(j )(1 ? sup P(Ln(j ) = x))5  A2 : n

x

j =1 p=1

Applying (2), this gives (5)

x

j =1

lim sup

d X

n!1 j =1

zn (j )(1 ? sup P(Ln(j ) = x))  A2 : x

Thus since zn (j ) ! 1, supx P(Ln (j ) = x) ! 1 as n ! 1. We can do better. Let ln be such that P(Ln (j ) = ln (j )) = sup P(Ln (j ) = x) for all j: x

From (5) we have that lim supn!1 zn (j )P(Ln (j ) 6= ln (j ))  A2 . So, in the limit as n ! 1, P(

zX (j ) n

p=1

Ln(p) (j ) = zn (j )ln (j ))  P(Ln(j ) = ln(j ))z

n

 (1 ? A2 =zn(j ))z ! e?A > 0:

n

(j )

(j )

2

As the LHS is bounded away from 0, we must have from (2), that for all n large enough (6)

d X n c = 0 zn (j )ln (j ): j =1

In particular, if ~y = fyn g1 n=0 is another realisation of Z0 (1) for which L0 (1)(~y ) = c, then for large enough n (7)

znT ln = ynT ln = 0nc =: an:

Now, let ?n = fZ0;n (1;
)ln = an g T = f~y = fyk g1 k=0 2 : yn ln = an g:

Clearly, showing (1) is equivalent to showing that limn!1 ?n \  exits and equals fL0 (1) = cg a.s. From (7), for any ~y 2 fL0 (1) = cg, we have ~y 2 ?n \  eventually, that is (8)

fL0 (1) = cg  lim inf ? \ : n!1 n

Continuity for branching processes

5

1 1 For ~y = fyn g1 n=0 2  we have fyn (j )gn=0  fzn (j )gn=0 for all j . Suppose that ~y 2 lim supn!1 ?n \  = f?n in nitely ofteng \ . That is, we have a sequence fn(k)g1k=0 such that n(k) ! 1 and ~y 2 ?n(k) \  for all k. Then, for any k,

n(k) (~y) = P(L0 (1) = cjZ0;n(k) (1;
) = yn(k) ) =

 

( ) (j ) d d yX X X n ( k ) Ln(p()k) (j ) = n0 (k) yn(k) (j )ln(k) (j )) P(0 j =1 j =1 p=1 d Y P(Ln(k) (j ) = ln(k) (j ))y ( ) (j ) j =1 Yd (1 ? A2 =zn(k) (j ))y ( ) (j ) from (5) j =1 n k

n k

n k

! C (~y) > 0 as k ! 1:

Thus as n ! 0 or 1, we must have from the above that n (~y) ! 1. That is, up to a set of measure 0, lim sup ?n \   fn ! 1g = fL0 (1) = cg: n!1

Combining this with (8) establishes (1). From (1) it follows that P(L0 (1) = c) = nlim !1 P(?n ; ?n+1 ; : : : ; ) = nlim !1 P(?n ; )P(?n+1 ; ?n+2 ; : : : j?n ; ) = nlim !1 P(?n ; )

1 Y

r=n

P(?r+1 j?r ; ) (Markov property)

= P(L0 (1) = c) nlim !1

1 Y r=n

P(?r+1 j?r ; ):

Q P(? j? ; ) = 1 and we have that Thus limn!1 1 r+1 r r=n (9)

1 X n=0

(1 ? P(?n+1 j?n ; )) < 1:

This is a necessary condition for there to be a jump in L0 (1) at some point c > 0.

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Owen Dafydd Jones

Let cn (?n \ ) = fyn : ~y = fyk g1 k=0 2 ?n \ g be the nth co-ordinate set of ?n \ , then P(?n+1 j?n ; ) =

X

y2c (? \) n



P(

n

y(j ) d X X j =1 p=1

Xn(p) (j;
)ln+1 = an+1 )

P(Z0;n (1;
) = yT jZ0;n(1;
)ln = an; )

X y2c (? \) n

n

n

n

0d 1? 21 X A @ y(j )(1 ? sup P(Xn (j;
)ln+1 = a))A a

j =1

P(Z0;n (1;
) = yT jZ0;n(1;
)ln = an; ) from (3) X pkyk A(1 ? u ) P(Z0;n(1;
) = yT jZ0;n(1;
)ln = an; )  1 n y2c (? \)  q

A

anklnk?1 1 (1 ? un)

:

The penultimate line requires ln+1 6= 0. The last line uses the fact that yT ln = an for all y in the sum. As P(Ln = ln ) ! 1, it follows that kln ? wn k1 ! 0, where wn = ELn. But kwn k1 = 1, so for all n large P(?n+1 j?n ; )  p 2A an(1 ? un) whence, for some nite constant B , 1 1 X X (1 ? P(?n+1 j?n ; ))  B + (1 ? p 2A ): an(1 ? un) n=0 n=0 So if the sum on the RHS is in nite, there can be no jump at c > 0. That is, a sucient condition for there to be no jumps in (0; 1) is that (10) lim sup p 2A < 1: n!1 an (1 ? un) This is equivalent to the condition lim sup(0n (1 ? un ))?1 < c=(4A2 ). However, as we must allow for any value of c 2 (0; 1), we eectively require 0n (1 ? un) ! 1: Note that, looking at where un appears in the proof, it is sucient when calculating un to evaluate supl=6 02R ;a2R P(Xn (i;
)l = a) only over l \close" to wn+1 . In particular, if the fwn g converge to some limit w, then if w(i) 6= 0 we can require l(i) 6= 0 for all large n. Note also that it can be shown that it is sucient to take l 2 Zd+ and a 2 Z+. d

+

+

Continuity for branching processes

7

3. Examples Clearly, the application of this result requires detailed knowledge of the distributions of the fXn g. In Jones (1997), a particular MTBPVE was examined, arising from the construction of a spatially inhomogeneous diusion on the Sierpinski gasket, a simple fractal. It is relatively straightforward, if somewhat tedious, to verify that the conditions of Proposition 2.1 hold in this case. It then follows immediately from the continuity of the branching process limit, that certain crossing times of the associated diusion are continuous on R, noting that it has previously been established that there is no jump at 0. To illustrate that the conditions of Proposition 2.1 are the sort of conditions we should be looking for, we present the following, simpler, example: Example 3.1 We consider a process with two types. De ne Xn by

x

(2; 0) (1; 1) (0; 2) (2; 2)

P(Xn (1;
) = x) = P(Xn (2;
) = x) 1 4 1 2 ?1 n 4 n

where n ! 0. Whence for n > m

Mm;n =

" nY ?1 k=m

#

2(1 + k )

1 21 2

1 12 2



:

Qn?1 m m Taking mRn (i) = 1T Mm;n (
; i), we get m n = Rn (1) = Rn (2) = k=m 2(1 + k ), and from Jones (1997) Theorems 1 and 2, a:s: L2

n0 Z0;n ???!

 1 L (1) 1 L (1)  0 21 0 : 12

2 L0 (2) 2 L0 (2) D L (2). Moreover, from the symmetry of the fXn g, L0 (1) = 0 D Z (2;
)1, then fT g1 is a (single-type) branching De ne T0;n = Z0;n (1;
)1 = 0;n 0;n n=0 process with varying environment, with ospring generation rules Yn given by

x P(Yn = x) 2 1 ? n : 4 n a:s: L2

D L (1); L (2). From Cohn (1996) Note that E T0;n = 0n and n0 T0;n ???! W0 (say) = 0 0 Theorem 1 we have that W0 is continuous on (0; 1) if and only if

(11)

1 X

n=1

2n n = 1:

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Owen Dafydd Jones

Compare this with Proposition 2.1,h which gives us i that L0 (1) and L0(2) are Q n ? 1 0 continuous on (0; 1) if n (1 ? un ) = k=0 2(1 + k ) n ! 1. This limit is nite for n = 2?n , but in nite if 2n n ! 1. So the conditions of Propostion 2.1 do miss some borderline cases, but are none-the-less very close to the sucient condition given by (11).

References [1] Asmussen, S. and Hering, H. (1983) Branching Processes. Volume 3 of Progress in Probability. Birkhauser. [2] Athreya, K.B. and Ney, P.E. (1972) Branching Processes. Springer-Verlag. [3] Cohn, H. (1996) On the asymptotic patterns of supercritical branching processes in varying environments. Ann. Appl. Prob. 6 896{902. [4] Cohn, H. and Jagers, P. (1994) General branching processes in varying environment. Ann. Appl. Prob. 4 184{193. [5] Cohn, H. and Nerman, O. (1990) On products of non-negative matrices. Ann. Prob. 18 1806{1815. [6] Grimmett, G.R. and Stirzaker, D.R. (1992) Probability and Random Processes. Second Edition. Clarendon Press, Oxford. [7] Jones, O.D. (1997) On the convergence of multi-type branching processes with varying environments. Ann. Appl. Prob. 7 772{801. [8] Petrov, V.V. (1975) Sums of Independent Random Variables. Springer-Verlag.