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Total Progeny for Systems Without Source

1. Returning to the simple problem without source, let Pk (i; jl, , t) be the probability that in the total progeny in all generations 1 through k produced by one particle of type i (generation 0), there should be ji particles of type 1,..., jt particles of type t. Define ci)(x) = Z Pk (i; jl, .jt) x^j '^..X-j and C(k)(x) as the corresponding transformation of It. Here the upper k does not indicate iteration. Clearly P(i; j) = pi(i; ), hence cl1)(x) = gi(x) and C(¹ ) = G(x).

Now let k be greater than 1. The production of the total state J,...,Jt at the end of the k-th generation from one particle of type i arises from the mutually exclusive states jl, - jt; 0 < jh < Jh in the first generation. If this state is 0, .. ., 0, then and only then will the total state J₁,. .., Jt be 0,... ,0, so Pk(i; O) = p(i; 0).

Suppose then that state J is not 0, and hence state j $ 0. Each of the jh particles of type h in the first generations acts independently of the others, and of those of other types to produce in the k - 1 next generations a total state of some al,...,at particles with probability Pk-1 (h; ai,...,at). We want the total state from the ji,... ,jt particles of the first generation to be J₁- ii,..., Jt - jt after the next k - 1 generations. It follows from the elementary laws of probability that, for J 0,

O<jk<Jkji jt Pk(i; J) = Zpi(i;j) ZfPk-l (1; al... at)...f JPk- (t; al,..., at) jiO E ai=Ji-ji But this is the coefficient of xiJ ... xtJt in gi (xic¹⁾ (x),.^. . ,XtCt^- )(X)⁾ = ^pl (i;^j)xjl . x^j t [ Pk-1 (1; a)xa ] . [ Pk-l(t; a)xa] and Pk_l(i;0) = pl(i;0), which is the constant term of the above function. Hence the

Theorem. The generating transformation C(k)(x) for the total progeny in generations 1 through k satisfies the recursive relations cM (x) = gi(x) ci (x) = gi (xc 1)(X),. ..,tC_t (x)

2. If y,z are arbitrary points of It, we have from Taylor's form, gi(Y) = gi(z) + E (gi/dxj)p (yj - zj) and thus gi(y) - gi(z) _I<mij lyj- zj \. It can be said therefore that the number dk _ ci (x) - c(x)I < Emij Ixjck _xjc -² 1 < mijd k-1 since is in It Iteration of this inequality yields dk < mrk^-2 dk-2, and eventually dk < Zm (k-2 d(2) = -En -2g9j (xG(x) - gj()) I<ZE,k-² Emjn <_ ijJ = Z ij _ ij Xn9n(X) -Xn= Z mi'n Xn 9n(X) - 1 <m(k¹ ) < Vrk1. n Thus we have the

Theorem. If G is below critical, the generating functions ck(x) are uniformly convergent to continuous limit functions ci(x) on It. The latter satisfy the functional equations ci(x) = gi (xicl(x), . . . ,xtt(x)).

3. We seek now a dominating sequence for Di-=Ocik/xj ck-1 aXI First, note that Pj -ac/Qxj = Z (OgiO/Xn)p [5nj Cn x]+XnC ¹ /j] <mij+ P\j , where P = xC^k- l. Iteration leads to P^1<mij + m m) pkj P2 and eventually pk <+m(2)+. . .+m (k-)+Em(k-)(gn/j)<mni+m( + (+mk) a~ <.~ij +.~ij ij in _j +j

For brevity, let Ak and Bn denote temporarily the round and square brackets involved in the Pj sum above. Then D= pk pk-P= |E aBk _ An¹ <AkIBk -B \k-1 + EB 1A- -A - < min,lBk -Bk-11' + E Bk-'llAk - Ak-117

We obtain upper bounds for the three A, B expressions: (1) Bk -B-'- =(cJ- I6ck-2)+ X (cC-1 _¹ /X- -c+² /iXj) < (2) Bn¹ < ⁶ nj + c a² /oxj<6nj + (mnj +..+ mj ) (3) IAk - A k-¹(Ogi/X.l)XCk-1 - (-g9il/Xn)Xk-2<a&gi/azXnpzlxpc 1 _pC-k-21 Bdk- 1 < BtVr ^k-2

Hence, combining, Dk < E min (6njVr^k-2 + Dj ) + BtVr^k-2 (6nj + mj +. +m( 2) < mijVr + E mi,nDn¹ + BtVrk^- + BtVr-²n(mj+. + m 2))j m ijVrk² tV+ BVkBtVr2 (I+ Bt + Wrk2) + E mn D - < mTijVr- ² + BtVr^-2 + BtVr-2Wr/(l - r) + E minDn Thus we have Dk < Krk + E minD4nj¹

Iteration leads to Dk<Krk^-2 + Krk-³min + Krk-4Emin) + ... + Kr Emk 3) + F-mik ² )D We obtain an upper bound for D² j = dc² /Oqxj - dc/9xjj = 9 (gn/ x,)G g,i+ E (agn/xp).x•gX/ -X ogn/0x= (09g /aXj ) ,g-1 + I(Ogn/axj),) - (09gn/ j) + | E (a9gn /xp).X pOgp/O/xj <m,j + B > \xpgp-xp + mnpmpj<m,j + +Bt

Therefore, substituting gives Dk < Krk² (k - 2) + E (-) + E (k-2)m + tm(2) < Krk-2(k - 2) + Vr^k- l + Vrk + Btrk^-2 . Since each of these terms defines a convergent series so does their sum and, we have

Theorem. If G is below-critical, the sequences ack/x₃j are uniformly convergent on It. Hence the partials Oci,/xj exist on It, and are the limit functions of the corresponding sequences. Since lim (Mcr/nxj) = (ci/xj), the latter is the limit approached by the expectation of particles of type j in the total progeny at the end of k generations from one particle of type i. From the functional equation satisfied by ci(x) follows En (mⁱ n- - in) (C,n/OXj)_I = -min, i = 1,...,t where IM -IIf 0, and the expectation limits are uniquely determined.