On a limit theorem for non-stationary branching processes.

Transcription

1 On a limit theorem for non-stationary branching processes. TETSUYA HATTORI an HIROSHI WATANABE 0. Introuction. The purpose of this paper is to give a limit theorem for a certain class of iscrete-time multi-type non-stationary branching processes (multi-type varying environment Galton-Watson processes). Let X N = t (X N,1,X N,2,,X N, ), N =0, 1, 2, be a iscrete-time branching process with types. The iscrete-time branching process X N is etermine by its generating functions φ n,n,i (z). For i {1, 2,,}, efine e i = t (e i,1,e i,2,,e i, ) Z + by e i,i =1, an e i,j =0, if i j.then (1) φ n,n,i (z) = α Z + ( i=1 z αi i ) Prob{X N = α X n = e i }, n Z +, N Z +, n N, i=1, 2,,, z C. The generating functions φ n,n = t (φ n,n,1,φ n,n,2,,φ n,n, ) are etermine by recursion relations (see, for example, [2] on branching processes). For a non-stationary branching process, we have (2) φ n,n (z) =φ n,n 1 (φ N 1,N (z)) = φ n,n+1 (φ n+1,n (z)), n < N, φ n,n (z) =z, Compare with the stationary case, φ n,n = φ 0,N n, or the one-type non-stationary case, =1, not many stuies seem to have been one for multi-type non-stationary branching processes. Perhaps one of the ifficulties with such stuies lies in fining a class of processes where the nonstationarity an multi-typeness enter in a non-trivial way, an yet a clear limiting behavior is obtaine. As an approach to this problem, we consier in this paper the case where the non-stationarity enters in a simple way. Namely, we assume (3) φ N 1,N (z) =D 1 N 1 F (D N z),n=1, 2, 3,, where D N = iag(d N,1,D N,2,,D N, ), N Z +, are iagonal matrices, an F is a C -value function in variables. From eq. (2) an eq. (3) it follows that φ n,n (z) =D 1 n F N n (D N z),

2 2 Non-stationary branching processes. which is similar to the corresponing formula for the stationary case. In fact, if the matrices D N are unit matrices, this equation reuces to the formula for the stationary case. If the limit ˆF (z) = lim N D 1 N 1 F (D Nz) exists an is analytic at z =1, then the asymptotic behavior of the branching process will be essentially that of a stationary branching process efine by φ N 1,N = ˆF. However, if ˆF is singular at z = 1, we cannot apply the theory of stationary branching processes irectly, hence a consieration of non-stationarity (N-epenences of φ N 1,N (z)) becomes non-trivial. We introuce some notation. Put 1 = t (1,, 1). For a C -value ifferentiable function F = t (F 1,F 2,,F )in variables z =(z 1,,z ), we efine a matrix F (z) by F (z) ij = F i(z). For a matrix z j M, M = sup Mv is the usual operator norm, with v=(v 1,,v ), v =1 v = v i 2 for a -component vector v. We also use the notation i=1 exp(t) = (exp(t 1 ),, exp(t )) for t =(t 1,,t ) C. Definition 1. Let {D N },N=0, 1, 2,, be a series of -imensional iagonal matrices, D N = iag(d N,1,D N,2,,D N, ), with positive iagonal elements; D N,i > 0. For a constant δ>1, we say that {D N } is of orer δ if D N,i < an δ N D 1 N,i < are satisfie. sup N Z +,i=1,, sup N Z +,i=1,, Definition 2. Let {D N }, N =0, 1, 2,, be a series of -imensional iagonal matrices of orer δ (> 1) an let l be a constant satisfying l>δ. We say that a C -value function F in variables efine on an open set containing {D N 1} is ({D N },l)-regular if the following are satisfie. N Z + 1. The function F has the expression F (D N 1+t) =D N 1 1+A N t + F N (t), N Z +, t < C 0 δ N, where A N is a matrix inepenent of t, an F N is analytic in t < C 0 δ N an satisfies F N (t) C 1 δ N t 2, t < C 0 δ N,N Z +, for some positive constants C 0 an C 1 inepenent of N an t. 2. There exists a matrix A such that A A N <C 2 δ N,

3 T. Hattori an H. Watanabe 3 where C 2 is a constant inepenent of N. The matrix A has an eigenvalue l, an the absolute values of eigenvalues of A other than l are less than l. Furthermore, each Joran cell with the eigenvalue l is one imensional. The conition 1 implies that F (D N 1) = D N 1 1,N=1, 2, 3,, A N = F (D N 1),N=1, 2, 3,, an the conition 2 implies that the limit (4) B n = lim N l N+n A n+1 A n+2 A N exists (see Appenix). The conitions 1 an 2 roughly state that for each N Z +, the function F is analytic in a circle of center D N 1 an raius O(δ N ) an that the first orer term of F in z ominates the higher orer terms. Note that the conitions o not exclue the possibility that the function F is singular at the point z 0 = lim D N 1. N Remark. We can exten our results to the case where the function F has N-epenences; i.e. with the replacement F F N in the above efinition an in eq. (3), Theorem 1 an 2 below still hol. Definition 3. Let {D N }, N =0, 1, 2,, be a series of -imensional iagonal matrices of orer δ (> 1). We call X N a branching process of type (, {D N },l), if X N is a iscrete-time branching process with types, an if the generating functions efine by eq. (1) satisfy the recursion equations eq. (2) an eq. (3) for some ({D N },l)-regular function F. The conitions in Definition 1 an Definition 2.2 together with l>δare an extension of the notion of supercriticality. Definition 2.1, on the other han, is a rather restrictive conition implying the existence of moments of X N. In this paper we prove the following. Theorem 1. Let X N be a branching process of type (, {D N },l). Then l N D 1 N X N N. converges weakly to a ranom variable Y =(Y 1,,Y ) as E[Y j X n = e i ]=l n ( Dn 1 B n )ij, i =1, 2,,, j =1, 2,,, n Z +.

4 4 Non-stationary branching processes. Example. Let X N be a branching process with 2 types such that the transition probability P N (α, β) =Prob{X N+1 = β X N = α}, α Z + 2, β Z + 2, N Z +, is etermine by P N ((1, 0), (2, 0)) = (1 ɛ N+1 )2 1 ɛ, N P N ((1, 0), (k, 2)) = 3(1 ɛ N+1 )k ɛ 2 N+1 25(1 ɛ N ), k =0, 1, 2,, P N ((1, 0),β)=0, otherwise, P N ((0, 1), (k, 2)) = (1 ɛ N+1 )k ɛ 2 N+1 ɛ N,k=0, 1, 2,, P N ((0, 1),β)=0, otherwise. Here, ɛ N, ɛ N, N =0, 1,, are positive constants, efine recursively by F (1 ɛ N+1,ɛ N+1) =(1 ɛ N,ɛ N ),N Z +, with given positive constants ɛ 0 an ɛ 0, satisfying ɛ 0 < 1 2 an ɛ 0 < 1 2, an F is efine by F (x, y) =(x y 2 1 x, y 2 1 x ). It is easy to see that X N is a branching process of type (2, {D N }, 4) with D N = iag(1 ɛ N,ɛ N). The conitions in the efinitions are satisfie with δ = 5 3. Also it follows that ɛ N = Cδ N (1 + O(δ N )) an ɛ N = Cδ N 1 (1+O(δ N )), where C is a constant. Theorem 1 therefore implies that 4 N ((1 ɛ N ) 1 X N,1,ɛ 1 N X N,2) converges weakly as N. Using the asymptotic form of ɛ N an ɛ N, it further follows that 4 N X N,1 an (12/5) N X N,2 converges, or in other wors, the average number of type 1 will grow at the rate of 4 per generation asymptotically, while that of type 2 will grow at the rate of 12/5 per generation. Note that though the birth rate of type 2 from type 1 isappears in the limit N, the birth affects the average growth rate significantly in the limit, or otherwise the average growth rate for type 1 woul have been 2 instea of l =4. This is because a birth of type 2 will in turn give birth 1 of orer of type 1 in average, which iverges as N is increase. ɛ N For the one-type case = 1, the branching process of type ( = 1, {D N },l) falls into the class consiere by Biggins an D Souza [3]. In this case, the conitions in Definition 1 an Definition 2.2 together with l>δimply the uniform supercriticality, while Definition 2.1 implies the

5 T. Hattori an H. Watanabe 5 ominance conition. (We learne this from Prof. Biggins.) A result in [3] then implies that (for = 1) the process has a single rate of growth. As is seen in the above example, the conitions (for >1) o not imply the same rate of growth among the ifferent types. The gap between the upper an lower boun of D N,i in Definition 1 is crucial for such a possibility. We are particularly intereste in the situation where growth rate of ifferent types iffer, an where the birth of less increasing types affects the growth rates of ominant types. Non-stationarity an multi-typeness both enter in a non-trivial way in this phenomena. The present stuy in the particular class of branching processes grew out of our attempt in constructing a non-self-similar iffusion on the Sierpinski gasket. We briefly iscuss the application in Section 3 (the etails, however, will be left for the future publication). Acknowlegement. The authors woul like to thank Prof. M. T. Barlow, Prof. K. Hattori, Prof. J. Kigami, Prof. T. Kumagai, Prof. S. Kusuoka, an Prof. Y. Takahashi for valuable iscussions concerning the present work. They woul also like to thank Prof. J. D. Biggins who kinly informe them of the connection between the present work an his results on onetype varying environment Galton-Watson processes. 1. Main results. Let {φ n,n (z)} be a sequence of C -value functions efine by the recursion relation eq. (2) an eq. (3) for a C -value ({D N },l)-regular function F. Put f n,n (s) = D n φ n,n (exp( l N D 1 N s)) = F N n (D N exp( l N D 1 N s)). Theorem 2. There exist positive constants ɛ an C such that, for every n Z + : (i) The function f n,n (s), N n, is analytic on s < ɛl n δ n an converges uniformly to an analytic function f n (s) on s < ɛl n δ n as N. (ii) The function f n (s) has the expression with the boun f n (s) =D n 1 l n B n s + f n (s), s < ɛl n δ n, f n (s) Cδ n l 2n s 2, s < ɛl n δ n.

6 6 Non-stationary branching processes. Theorem 1 is obtaine from Theorem 2 by a stanar argument. Proof of Theorem 1 assuming Theorem 2. Put C + = {s C R(s) 0}, For each n Z + an N Z +, satisfying n N, an for each i {1, 2,,}, let Q n,n,i be the istribution of l N D 1 N X N conitione by X n = e i, an let Φ n,n,i (s) = exp( s w) Q n,n,i (w) R = exp( s w) Q n,n,i (w), s C +, [0, ) be the Laplace transform (generating function) of Q n,n,i. Put Φ n,n = (Φ n,n,1,, Φ n,n, ). With eq. (1) we have Φ n,n (s) =φ n,n (exp( l N D 1 N s)). Theorem 2 then implies that there exists a positive constant ɛ such that {Φ n,n } N=n,n+1,n+2, converges uniformly to an analytic function on s < ɛ( ɛl n δ n ). Therefore we have for each i {1, 2,,} an each n Z +, M = sup{ exp(ɛ 1 w/(2 ))Q n,n,i (w)} <. N [0, ) Fix R>0 an put E j = {w [0, ) w j R},j=1, 2,,. Then we have M sup{ exp(ɛ 1 w/(2 ))Q n,n,i (w)} N E j exp(ɛr/(2 )) sup Q n,n,i (E j ). N Hence sup Q n,n,i ([0,R] ) 1 N j sup Q n,n,i (E j ) N 1 Mexp( ɛr/(2 )). The right han sie converges to 1 as R. Therefore the sequence of measures {Q n,n,i } N=n,n+1, is tight. Since {Q n,n,i } is tight, there exists a subsequence that converges weakly to a probability measure. For each convergent subsequence {Q n,kn,i} N=1,2,, {Φ n,kn,i(s)} converges uniformly on compact sets in

7 T. Hattori an H. Watanabe 7 C +. Therefore {Φ n,kn,i(s)} converges to a function that is analytic in the interior of C + an continuous in C +. On the other han, Theorem 2 implies that {Φ n,n,i (s)} converges to an analytic function in a neighborhoo of s =0. Therefore, {Φ n,kn,i} converges to an analytic function inepenent of the choice of subsequences in the interior of C +. Since the limit of {Φ n,kn,i} is continuous in C +, this further implies that {Φ n,kn,i} converges to a continuous function inepenent of the choice of subsequences in C +. We have shown that for any convergent subsequence {Q n,kn,i} of {Q n,n,i } the corresponing sequence of the characteristic functions {Φ n,kn,i( 1x)} converges to a function inepenent of the choice of subsequences for x R, which implies that the limit measure Q n,i of the sequence {Q n,kn,i} is inepenent of the subsequence. Since {Q n,n,i } is tight, this implies that {Q n,n,i } converges weakly to a probability measure Q n,i. The proof of the statement on the expectation values is straightforwar. This completes the proof. 2. Proof of Theorem 2. We put, for n 1,n 2 Z + with n 1 n 2, (5) (6) B n1,n 2 = l n 2+n 1 A n1+1a n1+2 A n2, B n1,n 1 = I. Theorem 2 follows from Propositions 1 an 2 below. Proposition 1. There exist positive constants ɛ an C 3 inepenent of n, N Z + such that the function f n,n (s), n N,has the expression (7) f n,n (s) =D n 1 l n B n,n s + f n,n (s), s < 2ɛl n δ n, with the boun (8) f n,n (s) C 3 δ n l 2n s 2, s < 2ɛl n δ n. Proposition 2. For n, N, N Z + with n N N, it hols that (9) f n,n (s) f n,n (s) C 4 r N s 2, s < ɛl n δ n, where C 4 > 0 an r>1 are constants inepenent of n, N, N an s. Proof of Theorem 2 assuming Propositions 1 an 2. Proposition 2 implies that the family { f n,n } N=n,n+1, constitutes a Cauchy series. Then, taking the limit N in (7) an (8), we obtain Theorem 2, where f n (s) = lim f n,n (s). N

8 8 Non-stationary branching processes. Proof of Proposition 1. Let us prove the proposition by inuction on n = N,N 1,, 0. Since (10) f N,N (s) =D N exp( l N D 1 N s) an since there exists a constant C 5 > 0 such that we have l N D 1 N s < C 5, s < 2ɛl N δ N, (11) f N,N (s) αδ N l 2N s 2, s < 2ɛl N δ N, for some constant α inepenent of N an s. Suppose that (12) f n,n (s) α n δ n l 2n s 2, s < 2ɛl n δ n, hols for an n N, where α n is a constant inepenent of N an s. Let us estimate the function f n 1,N (s) given by (13) f n 1,N (s) =F (f n,n (s)) = F (D n 1 l n B n,n s + f n,n (s)). Firstly, the function f n 1,N (s) is well-efine by (13) for s such that (14) l n B n,n s f n,n (s) < C 0 δ n. As is shown in Appenix, there exists a constant C 6 > 0 inepenent of n such that (15) B n,n <C 6. Then the conition (14) hols for s satisfying (16) s < 2ɛl n 1 δ n+1, if (17) C 6 δ l 2ɛ +4ɛ2 ( δ l )2 α n <C 0. Seconly, by using f n 1,N (s) =A n fn,n (s)+ F n ( l n B n,n s + f n,n (s))

9 T. Hattori an H. Watanabe 9 an l n B n,n s f n,n (s) (C 6 +2ɛα n )l n s, we can estimate f n 1,N (s) an reprouce (12) with n replace by n 1, where we put (18) α n 1 = A n δl 2 α n + C 1 δl 2 (C 6 +2ɛα n ) 2. Since δ<lan A n <l+ C 2 δ n, the solution to (18) with α N = α satisfies (17), if ɛ is sufficiently small. The proof of Proposition 2 is base on the recursion relation (19) f m 1,N (s) f m 1,N (s) =F (f m,n (s)) F (f m,n (s)) an on the following ifference property for F. Lemma. For m Z + an t, t C such that t, t < C0 2 δ m, (20) F (D m 1+t) F (D m 1+t )=(A m + G m )(t t ), where G m = G m (t, t ) obeys the boun (21) G m C 7 δ m max( t, t ). Proof. The matrix element of A m + G m is given by the corresponing element of F at some point on the line segment connecting D m 1+t an D m 1+t. Note that F (D m 1+t )=A m + F m (D m 1+t ) hols an that F m (D m 1+t ) is analytic on {t C t <C 0 δ m } an boune by C 1 δ m t 2. Now, we fix t such that t < C0 2 δ m an efine the function v(τ) = F m (D m 1+t + τe i )ofτ analytic on {τ C τ < C0 2 δ m }, where e i = t (e i,1,e i,2,,e i, ), e i,j = δ i,j.

10 10 Non-stationary branching processes. Then, integrating the function v(τ)τ 2 along the contour τ = t,we can boun v (0) = F m (D m 1+t ). As a result, we have t i F t m (D m 1+t v(τ) ) sup i τ = t t 4C 1 δ m t, t < C 0 2 δ m. This proves the lemma. Proof of Proposition 2. Fixing n, N, N an s such that n N N an s < ɛδ n l n, we estimate f m,n (s) f m,n (s),m= N,N 1,,n. As is shown in Appenix, there exist positive constants C 8,r > 0 with 1 <r <lsuch that (22) B m,n B m,n < C 8 r N+m, m N<N. Now, (7), (19), an (20) imply (23) f m 1,N (s) f m 1,N (s) =F (f m,n (s)) F (f m,n (s)) + l m+1 (B m 1,N B m 1,N )s =(A m + G m )(f m,n (s) f m,n (s)) + l m+1 (B m 1,N B m 1,N )s =(A m + G m )( f m,n (s) f m,n (s)) + G m l m (B m,n B m,n )s, where G m epens on s. By the argument obtaining (14) from (15), (16), an (17) with the ɛ replace by ɛ 2, we see that t an t efine by t = D m 1+f m,n (s) =l m B m,n s f m,n (s) t = D m 1+f m,n (s) =l m B m,n s f m,n (s) satisfy the assumption of Lemma. Then we have the boun since (8) an (15) imply G m C 7 (C 6 + C 3 ɛ)δ m l m s, t, t (C 6 + C 3 ɛ)l m s. Then, from (23), (15), an (22), we obtain (24) f m 1,N (s) f m 1,N (s) A m + G m f m,n (s) f m,n (s) + C 7 C 8 (C 6 + C 3 ɛ)δ m l 2m r N+m s 2.

11 On the other han, (8) implies T. Hattori an H. Watanabe 11 (25) f N,N (s) f N,N (s) f N,N (s) + f N,N (s) 2C 3 δ N l 2N s 2. The inequalities (24) an (25) together with the boun A m + G m <l+ C 2 δ m + C 7 (C 6 + C 3 ɛ)( δ l )m s <l+ C 2 δ m + C 7 ɛ(c 6 + C 3 ɛ)( δ l )m n prove (9) with r = min(r, l/δ). 3. An application. In this section, we briefly explain an application of our theorems to the construction of asymptotically one-imensional iffusion processes on fractals. (The iea is announce in [5]. Details are in preparation.) Unlike the other part of this paper, we assume in this section that the reaer is familiar with the works on iffusion processes on finitely ramifie fractals (for example, [1][4][7][8][9]). In the construction of iffusion processes on the finitely ramifie fractals, one starts with a set of ranom walks on pre-fractals an obtains the iffusion process on the fractal as a weak limit of the (time-scale) ranom walks. This approach was first establishe for the Sierpinski gasket [4][8][1]. One of the keys to this construction is ecimation invariance, which means that the transition probability of the ranom walk is a fixe point of the corresponing renormalization group transformations (in the terminology of [6;section4.9]) in the space of transition probabilities. In general, one may assign ifferent transition probabilities to ifferent kins of jumps of the ranom walk, in which case the space of transition probabilities become multi-imensional. The conition that the obtaine iffusion process spans the whole fractal implies that every component of the fixe point is positive (non-egenerate fixe point). In fact, Linstrøm efine a class of fractals (neste fractals) in such a way that the corresponing renormalization group has a non-egenerate fixe point [9;section IV, V], an succeee in constructing iffusion processes on neste fractals. In [6;section 5.4] we introuce abc-gaskets as examples of the fractals where non-egenerate fixe points of the corresponing renormalization groups are absent (if the parameters a, b, c satisfy certain conitions), while they always have unstable egenerate fixe points which correspon to the ranom walks on (one-imensional) chains. The problem then arises; can we construct iffusion processes on finitely ramifie fractals whose renormalization groups have only egenerate fixe points?

12 12 Non-stationary branching processes. The iea of the solution is to use the renormalization group trajectories that emerge from a neighborhoo of the unstable egenerate fixe points. We take the 111-gasket (which is just the Sierpinski gasket), as an example to explain briefly our iea of the construction, with emphasis on how the problem is relate to non-stationary branching processes. Let G n enote the set of vertices of the pre-sierpinski gasket with the smallest unit being the equilateral triangle of sie length 2 n (see, for example, [1] for the notation), an let G = G n enote the Sierpinski gasket. For a process X taking values in G we efine a stopping time Ti n(x), n =0, 1, 2,, by T 0 n (X) = inf{t 0 X(t) G n }, an Ti+1 n (X) = inf{t >Tn i (X) X(t) G n \{X(Ti n (X))}} for i =0, 1, 2,. Ti n(x) is the time that X hits G n for the i-th time, counting only once if it hits the same point more than once in a row. Denote by W n (X) = T1 n(x) T 0 n(x) the time interval to hit two points in G n. For an integer n an a process X on G or on G N for some N > n, ecimation is an operation that assigns a walk X on G n efine by X (i) =X(Ti n(x)). The constructions in [4][8][1] use the sequence of simple ranom walks; ranom walks with same transition probability in every irection. The sequence of simple ranom walks {Y N } on G N (N =1, 2, 3 ) has the property of ecimation invariance; namely, the ranom walk Y on G n efine by Y (i) =Y N (Ti n(y N )) has the same law as Y n. This implies that {W n (Y N )},N= n +1,n+2,n+3,, is a (one-type) stationary branching process (see, for example, [1;Lemma2.5(b)]). A limit theorem for a stationary branching process gives a necessary estimate, which, together with other ingreients, finally leas to the theorem that the sequence of processes X N, N =1, 2, 3,, efine by X N (t) =Y N ([λ N t]), where λ = E[W 0 (Y 1 )], converges weakly to a iffusion process as N. The situation is similar for the case of multi-imensional parameter space, which appears in neste fractals [8], where the limit theorems for multi-type stationary branching processes can be employe for necessary estimates [7]. As a generalization of the simple ranom walk let us consier a ranom walk Z = Z N, x, x = t (x, y, z), on G N efine as follows. Z is a Markov chain taking values on G N, an the transition probability is efine in such a way that at each integer time Z jumps to one of the four neighboring points an the relative rate of the jump is x : y : z for {a horizontal jump} : {a jump in 60 (or 120 ) irection} : {a jump in 120 (or 60 ) irection}. For a simple ranom walk, x = y = z. The parameter space P can be efine as P = {(x, y, z) x + y + z = 1,x 0,y 0,z 0}. There is a one to one corresponence between a point in P an a ranom walk. Z N, x oes not have the property of ecimation invariance. Instea, the ecimate walk Z efine by Z (i) =Z N, x (Ti N 1 (Z N, x )) has the same law as Z N 1, x, where x = T x = t (C (x + yz/3), C(y + zx/3), C(z + n=0

13 T. Hattori an H. Watanabe 13 xy/3)), 1/C =1+(xy + yz + zx)/3. The map T maps P into P.The ynamical system on the parameter space P efine by T is the renormalization group. The simple ranom walk correspons to (1/3, 1/3, 1/3) P which is a non-egenerate fixe point of T.(0, 0, 1) P is a egenerate fixe point corresponing to the ranom walk along a one-imensional chain. We choose a sequence {Z N, xn } as follows. Let 0 < w 0 < 1 an efine w N, N = 1, 2, 3,, inuctively by w N+1 = (6 w N ) ( 2+3w 1 N + ( ) ) 4+6w N +6wN 2 1/2. Note that w 0 >w 1 >w 2 > 0. Put x N =(1+2w N ) 1 (w N,w N, 1) an consier Z N = Z N, xn. One then sees that if n<n, then Z n is a ecimation of Z N. This property of {Z N } correspons to the ecimation invariance of simple ranom walks {Y N }. The special choice of the parameters simply means x N 1 = T x N. Letting N, x N approaches the egenerate unstable fixe point (0, 0, 1). Put W1 n(z N)=W n (Z N ), an let W2 n(z N ) be the number of iagonal (off-horizontal) jumps in the time interval (T0 n (Z N ), T1 n (Z N )], an W3 n(z N ) be the number of visit in the same time interval to the points from which two horizontal lines emerges. Let D N,N=0, ( 1, 2,, be a sequence ) 1 1+3w N of iagonal matrices efine by D N = iag,w N,. 1+3w N 2+2w N Then by explicit calculations one obtains the following Proposition. Proposition 3. W N = t (W n 1 (Z n+n ), W n 2 (Z n+n ), W n 3 (Z n+n )), N = 0, 1, 2,, is a branching process of type ( =3, {D N },l=6). With this proposition, Theorems 1 an 2 can be applie, which take a part of the role of the limit theorems for stationary branching processes for the fixe point theories. Together with other ingreients we finally obtain the theorem that the sequence of processes X N, N =1, 2, 3,, efine by X N (t) =Z N ([λ N t]), where λ = l, converges weakly to a iffusion process as N. (The resulting process is ifferent from the alreay known iffusion process on the Sierpinski gasket.) A Proposition similar to the Proposition 3 hols for general abc-gaskets, hence the Theorems 1 an 2 are generally applicable. In contrast to the ecimation invariant (fixe point) theories, where the existing limit theorems for the stationary branching processes worke, we neee limit theorems for the non-stationary multi-type branching processes. The present stuy grew out of such problems. Appenix. In what follows, we show that the matrix B n,n efine by (5) an (6) satisfies (15) an (22) for some constants C 6,C 8,r > 0 with 1 <r <l uner the assumption 2 in Definition 2. The matrix B n efine by (4) exists if (22) hols.

14 14 Non-stationary branching processes. Proof of (15). Since we have l 1 A k l 1 A +l 1 A A k < 1+l 1 C 2 δ k, B n,n < Proof of (22). Since we have Then B n1,n 2 (l 1 A) n2 n1 = N k=n+1 n 2 1 B n1,n 2 (l 1 A) n2 n1 < (1 + l 1 C 2 δ k ) <C 6. k=n 1 B n1,kl 1 (A k+1 A)(l 1 A) n2 k 1, n 2 1 k=n 1 C 6 l 1 C 2 δ k 1 <C 9 δ n1. B k,n B k,n B k,n (l 1 A) N k + B k,n (l 1 A) N k + (l 1 A) N k (1 (l 1 A) N N ) < 2C 9 δ k + C 10 ( l 1 l )N k, where l 1 <lis a positive constant such that the absolute values of eigenvalues of the matrix A except l are less than l 1. Therefore we have, for m k N, B m,n B m,n B m,k B k,n B k,n <C 6 (2C 9 δ k + C 10 ( l 1 l )N k ). We here assume that N m is even without loss of generality an put k =(N + m)/2. Then the above estimate turns out to be B m,n B m,n < C 6 (2C 9 δ N +m 2 + C 10 ( l 1 l ) N m 2 ).

15 T. Hattori an H. Watanabe 15 Choosing r = min(δ 1/2, (l/l 1 ) 1/2 ), we obtain (22). Obviously, 1 <r <l hols. REFERENCES [1] M. T. Barlow, E. A. Perkins, Brownian Motion on the Sierpinski Gasket, Probab. Th. Rel. Fiels 79 (1988) [2] B. A. Cevaschanov, Branching Processes, Nauka, Moscow, [3] J. C. D Souza, J.D. Biggins, The Supercritical Galton-Watson Process in Varying Environments on one-type varying environment Galton- Watson process, Stoc. Proc. Appl. 42 (1992). [4] S. Golstein, Ranom walks an iffusion on fractals, IMA Math. Appl. 8 (1987) [5] T. Hattori, Construction of asymptotically one-imensional continuous Markov process on Sierpinski gasket, RIMS Kokyuroku 783 (1992) (in Japanese). [6] K. Hattori, T. Hattori, H. Watanabe, Gaussian fiel theories on general networks an the spectral imensions, Progr. Theor. Phys. Supplement 92 (1987) [7] T. Kumagai, Estimates of the transition ensities for Brownian motion on neste fractals, To appear in Probab. Th. Rel. Fiels. [8] S. Kusuoka, A iffusion process on a fractal, Proc. Taniguchi Symp. (1987) [9] T. Linstrøm, Brownian motion on neste fractals, Mem. Amer. Math. Soc. 420 (1990) Tetsuya Hattori Department of Information Sciences Faculty of Engineering Utsunomiya University Ishii-cho Utsunomiya 321 JAPAN Hiroshi Watanabe Department of Mathematics Nippon Meical School Nakahara-ku Kawasaki 211 JAPAN