II

Abstract

We continue in this report the generalization of the methods and results of Hawkins and Ulam² which we began in I, being concerned principally with systems which are below critical. After deriving necessary and sufficient conditions for this state in terms of first moments, we study the direction of flow induced in the "unit cube" by the corresponding generating transformation. The latter results are used to show that the distribution of the generation in which death first occurs possesses moments of all orders.

Limits of expectations are obtained for the problem of subcritical system with source, and for that of total progeny (corpses) in subcritical systems.

Finally, it is shown how fictitious particles of new types may be introduced in such a way that certain more complicated problems may be reduced to the case of simple iteration of generating transformations. In particular we have shown how this may be accomplished for the system with source, and for the problem of total progeny.

The next section, III of this series, will deal with measure theorems on the space of all genealogies possible in multiplicative systems.

I—
Some Properties of the Jacobian

1. We consider a multiplicative system involving t types of particles, in which a particle of type i has a fixed probability pi(i;jl, .. jt) of producing a total of jl +... +jt particles, j, of type v, upon transformation. The corresponding generating transformation G(x) of the unit cube It:gi(x)- =-Pi (i; il,..*,jt) xⁱ '*6 xtit5

has the property that, upon iteration k times, the resulting transformation Gk(x): gi)() = Pk (i;jl,.. .,jt)x...xt has as its coefficient pk(i;j) the probability that the state (jl,...,jt) shall exist in the k-th generation of progeny from one particle of type i.

2. The transformation G(x) has for its Jacobian at x = 1 the first moment matrix J(G(1)) = [Ogi/axj = [mij][m,i j and, more generally, J(Gk(l)) = [gk)/axj] =[^mij ] where mk) is the expectation of particles of type j in the k-th generation of progeny from one particle of type i. Under our assumptions on G (see I), all these moments are positive. Moreover, we have seen that [mij] = [mij]

3. The importance of matrices with positive elements required study of their properties. We found that for such a matrix M, there is one and only one solution, r, v of the relations vM = rv, r > O, v > 0 (i.e., all vi > 0) In similar fashion one shows the existence and uniqueness of r', v' such that Mv' =r'v', r' > 0, v' > . That r = r' in these equations is evident from the following

Lemma. If vM = rv, r > 0, v > 0, and R is an arbitrary positive characteristic root of M, then r > R. Similarly, Mv' = r'v', r' > 0, v' >0, IM - RI_I=, R > 0 implies r'>R.

Proof. First statement: Let wM = Rw, w 5 0, and define b= min{wi/vi}, B = max{wi/vi}. Since all vi > O, bvi<wi <Bvi and hence, bVm(jk) <Em < BVim(k) and brkvj <RkW<BrkVj, all j. Since r > O, also bvj<(R/r)kwj<Bvj. Suppose R/r > 1. If at least one wj > 0 the right member yields a contradiction for large k. If at least one wj < 0 then the left member does. Hence all wj = 0, contradicting choice of w. Thus R/r< 1.

The second statement of the lemma is proved similarly.

We include for later use the trivial remark: If M is a matrix with positive elements, and Mv' = rv', r > O, v' > O, we have Mnv' = r"v^' for all positive integers n, and thus rnvj = m > min(vl ). Hence . m7(n) < r~v'/ min (v) -= rVi < rr max(vi) Vr", where V is a positive constant.

Similarly, if vM = ri, r > 0, v > 0, we have Ei m () < Wrn, where W is a positive constant.

4. We shall also need the following

Theorem. If M is a matrix of positive elements with Mv' = rv', r > 0, v' > 0, and Tk(v) is the transformation Mkv/s (Mkv), then limTk(v) = v' uniformly for all v $ 0, vi> 0.

Here s(w) indicates the sum E wi of the components of the vector w. The proof is entirely analogous to that used in I to prove the same result for row vectors.

5. We proved in I that lim Gk(O) = x^° exists: lin gk)(0) = limpk(i; 0)=X_i, and defined G as supercritical in case all x°< 1. Under our conditions on G, the alternative case is that all x = 1, hence x°= 1, and this case we called subcritical. Most of the present report will be devoted to systems of the latter type, in which the probability of death in generation k rises to limit one.

6. We have seen in I that a system is subcritical if and only if the maximal root r of the first moment matrix M = J(G(1)) is less than or equal to one. Equivalently G is subcritical if and only if the determinants IAnl of the upper principal minors An = [mij - Oij]₁ of the matrix A = [mij - ijl = M - I satisfy the relations: (-1)n+1lA1l < 0, n = 1,... ,t-1, (-1)t+lIM - I < .

7. We say a system is just-critical in case it is subcritical and the maximum positive characteristic root r of M is equal to one. A subcritical system with r < 1 is said to be below-critical. The justcritical case, while of theoretical interest is refractory, and we have limited ourselves for the most part to systems which are below-critical.

Theorem. A subcritical system is just-critical if and only if IM-II = 0. If r = 1, then of course 0 = \M-rI = IM-II, r being a characteristic root of M. Conversely, if IM - II = O, r > 1 by maximality of r, and r< 1 by assumption of subcriticality; hence r = 1.

8. We include for future reference the trivial remark: If M is a matrix of positive elements with Mv' = rv', 1 > r > O, v' > O, then there exists an e > 0 such that all vectors w in the e-neighborhood of v' are positive and satisfy the inequalities mi Wj < -(1 + r)Wi < Wi .

It suffices to note that the functions Ri(w) =_ mijwj/wi are continuous at w = v' and there have value Ri(v') = r < 1. Hence there is an e > 0 so that whenever llw - v'll< e, w will have positive components and Ri(w)< 1/2(1+ r).

II—
Direction of Flow of G[superscript(k)](x) in Subcritical Systems

1. In this section we study properties of the vector 1 - GK(x) with i-th component I-g?(k) (x), for x 1 in It, for a subcritical system, and of the vector Gk+l(x) = Gk(x) in a system below-critical. These results are of preliminary character, and are exploited in III.

Note that, for x $ 1 in It, we have x; I and thence gk)(x) < (k) (1) = 1 for all k so that the vector 1 - Gk(x) is not zero; in fact all its components are positive.

2. Recall that the e-cone of the vector v' consists of all vectors w such that (lw/a - v'll < e for some real positive a. We prove the

Theorem. If G is subcritical and x $ 1 is in It, then for every e > 0 there is a K so that for all k>K the vector I - Gk(x) is in the e-cone of v'.

Here v' denotes the characteristic vector of the relation Mv' rv', r > 0, v' > 0, where M is the first moment matrix of G. The theorem asserts, in geometric terms, that the direction from Gk(x) to 1 approaches the direction of v' with increasing k.

Proof. Fix x $ 1 in It, and e > 0. There exists a k (hereafter fixed) such that ITk(v) - v'\ < e/2Vt for all v $ 0, v_i> O. (The transformation Tk is that defined in I 4.) Since limGn"() = I and the first partials of Gk are continuous at 1, there exists an N such that for, all n> N, mre) = (rga/nXj) < eMk/2 (() where Mk min { E mk;ij j}, and the partial is evaluated at the point Gn (x)e. Fix n >N and define an = E (1-g)(i)) =s(Mk (1-G(x))). Then vi-( - (1-g x)/a[_va-y'(O-gk)/9xj)p (l-g)( n))/aI from Taylor's form of gk)(x) expanded about 1, and evaluated at x = Gx(z). Thus G(x)-<P-< 1.

The latter absolute value does not exceed v -Em (k)-g(n(- ))/a + ' -T(k) (1 - G(n)()) I +E(m()-_ (ag(k)/D X)) (1-gn)(i))/a~ < e/2vt + eMk/2Vt s (1 - C(n)()) /a. But ak> Mks (1 - C(n)(x)) so finally the original absolute value is seen to be less than e/vt. But then Ilv' - (1 - Gk+n(X)) /akll< e.

3. In the below-critical case in one variable (t = 1) the graph of the generating function g(x) is monotone increasing and concave up on the interval (0,1) with 1 = g(1). Moreover, the Jacobian J (g(1)) = g'(1) and hence the characteristic root r is g'(1), which is therefore less than one when g is below-critical. It is obvious geometrically therefore that for every x satisfying 0 <x < 1, the sequence of iterates gk(x) is monotone increasing: x< g(x) < g² (x) < .... This simple situation need not obtain in case t > 2. For purposes of illustration we regard the following example.

Consider the transformation G(x) of the unit square I2 defined by I I 1 g₁(x) = - + - 1X+ x1x21 1192(x)= - + -X2 + xx2 Computation of the four first partials at 1 shows that the first moment matrix is M= 4 I and thus M-I=1_41 The upper minor determinants of the latter satisfy (-1)² AI1 = -- < 0, (-1)3 A21 =-16 ^< 0. Hence G is below-critical. Indeed the characteristic equation of M is x² - x + 3/16 = 0 with roots 1/4 and 3/4. Thus r = 3/4 < 1.

(The right hand characteristic vector v' is found to be [ upon setting r = 3/4 in the matrix-vector equation (M - rI)v' = 0 and solving the resulting two homogeneous equations in v{,vs.)

Note that (see I), M being subcritical, we must have mij vj<vi for at least one i = 1, 2, whenever v # O, vi> 0, but not necessarily for both indices. In our case, for example, [14 [] [] where 9 < 8 but 3 > 2. 42 2 2

It is essentially this fact that causes the simplicity of the one variable case to break down. Specifically, let x = (4/5, 1/5). Then our G(x) at this point is (74/100, 59/100) and x -< G(x) is false for this x.

We can however prove the following

Lemma. If G is below-critical, and x $ 1 is in It, the k-sequences 9i(k)(x) are eventually monotone increasing.

Proof. By I 8, there is an e > 0 such that ilw - v'll < e implies w positive and mijwj< wi, all i.

By the preceding theorem, this e determines a K so that, for all k>K, II (1- Gk(x)) /ak - v'll < e. For all such k therefore, Zmij(1 -g(k)())/ak< (1 -g)())/ak and the positive ak may be deleted from this inequality. Under our assumptions on G, gi(x) > I + Emij(xj- 1) for every x with all components less than one, and thus I - gi(x) < Emij (- xj). In this inequality we may set x =Gk(x), since we have already shown that the latter enjoys this property of components (cf. II 1).

Thus 1 -g (k+l) ) < Em (1 - (k)(x)), and combining with the previous inequality, 1 -g^x )) < 1-g(k)x), whence the result desired.

4. We have seen that the direction of the vector 1 - Gk(x) approaches that of v'. We intend to prove the same result for the "vector of flow" Gk+l(x) - Gk(), with x 1 in It.

It is trivial that the latter vector is never zero for any k, and hence defines a direction. For otherwise, G would have a fixed point G^k () 1. Moreover, we know from the preceding result that for all k >K, all components of this vector are positive.

Theorem. If G is below-critical and x : 1 is in It, the direction from Gk(x) to Gk+l(x) approaches that of v'.

Proof is entirely analogous to that for 1 - Gk(x). Fix e > 0 and 2 5 1 in It. Then we may fix k so IITk(v) - v'le/2v/t for all v O 0, vl > 0, and next determine N so n >N implies the inequality (*) of II 2, and i(n)(2X) > g9i(). Now define An = k ij m=(k) (gk+n+l)(,) - g(k+)()) > Mk (Gk+n+l()- Gk+n(r)). Then Iv - (gi++x)( -gki ))/AVil(c (k)/Xj) p(g n+)( -jn)(2 ))/Ak < - ' /g)MJ)-J))/ by Taylor's theorem where Gn(x) -P < G"+L().

The remainder of the proof now proceeds just as in II 2. The essential point is that the Tk operates now on the vector Gk+n+l(p) Gk+n(z) which we know by II 3 to be positive and hence subject to the inequality v' - Tk (Gn+() - Gn(x)) l < e/2V. Thus the final result is Iv'- (Gk+'+n() - Gk+n()) /A\ < e for all n > N.

5. We now have immediately the main result which we want in III.

Theorem. If G is below-critical, and its first moment matrix satisfies the relation Mv' = rv', 1 > r > 0, v' > 0, and further, if x$ 1 is in It, then there is a K such that, for all k > K, the ratio of successive terms of the K-sequence gi +)() - gi()() is less than 1/2(1 + r).

From I 8, there is an e > 0 such that lw - v'll < e implies w positive and mnijwj < 1/2(1 + r)wi. This e determines K by the preceding theorem so that k>K implies

Gl(x)G() v <e Ak Hence EJ((k+(l)k(1)-9(k) ()) < i( + r) (k+)()- g(k)( )) since the positive Ak may be deleted. But (k+2) - _g(k) )=gi (G+ )) - gi(Gk(x)) = 9(gi/ x,)p((kgl)x) (k)(-g ))< Emij(gk)x) g(k)()) Hence, combining, we have the desired result.

III—
On the Distribution of Death in Subcritical Systems

1. Let G be subcritical, and define qk(i) as the probability that complete death of the system of progeny from one particle of type i should first occur in the k-th generation.

Clearly pk(i; 0), the probability of death in the k-th generation, may be expressed as k pk(i;) =qj(i) j=l Hence we have the relations: qk(i) = Pk(i; 0) - Pk-l(i; 0) =- gk)( ₀) - (k1(₀), k > 2, ql (i = p(i; 0) = gi(0)

2. We recall that x z x' implies gi(x) < gi(x') under our assumptions on G. Now 0 ~ G(0), otherwise G(0) = 0 and G would have a fixed point in It besides 1 and would be supercritical. Hence, inductively gi(O) < 92)(0 ) < g ( ()(0) < ..

It follows that the qk(i) are positive for k> 2, and ql(i) = gi(O) is positive for at least one i. Also, Z'qj(i) = limk Ek qj(i) = limk pk(i; 0) = 1 for every i, and thus the sequence ql(i), q ₂(i), q ₃(i),... is a probability density function on the positive integers.

3.Theorem. If G is below-critical, its first moment matrix M having maximal positive root r < 1, there is a K so that for all k >K, qk+l(i)/qk(i) < 1/2(1 + r). Consequently the density sequence qk(i) is eventually monotone decreasing, and all its moments ms(i)- k^sqk (i), S0 k exist.

The first statement is an immediate consequence of II 5, with x = 0. The finiteness of the moments follows from the ratio test: (k + 1)^sqk+l/kSqk < (1 + l/k)s (1 + r) - (1 +r) < .

4. In the case of a system below-critical in one variable (t = 1), it is geometrically obvious that qk+l/qk = gk+ (0) gk (0)/gk (0) gk-() = g(x') - g(x)/x' - x < g'(1) = r < 1 for all k, so qk < qlr^k- l and mi = Ekqk < q (1 + 2r + 3r²⁺ ...) =^{ql(l +x+x+...)} X = q₁ ((1- x)-)xr = ql/(l + r)^2. Hence in this case m₁< p(0)/(1 + r)2 = g(0)/(1 + g(1))2

5. Examples show that, even in the one variable case if the system is just-critical, even ml may be infinite. We hope to study the one variable case more completely in a separate report.

IV—
Subcritical System with Source

1. Consider t types of particles whose probabilities of transmutation are given by the generating transformation G(x): gi(x) =p(i;jil,...,jt)x^j l ...x^j as before. Suppose further that we have a source which emits independently into the system n+...+nt particles, ni of type i, with probability s(nl,...,nt)> 0. We associate with the source the generating function S(x)= s (nl,...,nt)xL ... xt, S(1) = 1

Consider a process consisting of the following steps:

1. The source produces an initial set of n, + ... + nt particles, ni of type i, with probability s(nl,...,nt). These particles transmute according to the G law to form a system which we regard as the first population.

2. The source again contributes new particles, and these together with the first population transform according to the G law to form the second population, and so on. At the k-th step, the population (ml,...,mt),mi of type i, will occur with some probability hk (ml,...,mt), and we define the corresponding generating function Hk(x) = , hk (ml,. ., m) x ...x^t

Now, from the elementary laws of probability, as we have pointed out in I, transmutation of any population with generating function N(x) according to the G law gives a population with generating function N (G(x)) = N (g ₁(x),...,gt(x)). Hence for the problem considered above, we see that Hl(x) = S(G(x)), H₂(x) = S (G(x))S (G² ()),..., and, generally Hk(x) = S (G(x)) Hk- ₁(G(x)) = S (G(x)) S (G2(x)) ... S (Gk (x)).

For, if Hk-i(x) is the generating function for the (k-1)st population, then the generating function for the intermediate population resulting from the contribution of the source to the (k - )st is Tk(x) = S(x) . Hk-1(x), and, upon transformation of this result by the G law, we must have for the generating function for the k-th population: Hk(x) = Tk (G(x)) = S (G(x)) - Hk-(G(x)) .

k 2. From Hk(x) = S (Gi(x)) follows Hk () = Hk- (x) . S (Gk ())

Since, for x in It, the latter factor is less than or equal to one, the k sequence Hk(x) is monotone non-increasing at x and H(x) - limHk(x) exists in It. Clearly 0 <H(x)< I and H(1) = 1.

If S satisfies the conditions

(S*) at least one OS/Oxj > 0 for all x : 0 on It, then for every x' with all components less than one, we have S(x') < 1. For S(x')1+ E (OS/9xj)p (x' - 1), where 0 ¢ P.

But for every x $ 1 in It, we have seen x' = g\)(x) < 1 for all i,k. Hence, for such x, S(x') = S (Gk(x)) < 1 and 1 > H₁(x) > H₂ (x) > ... so H(x) < 1. Thus we have

Theorem. The function H(x) - limHk(x) exists for all x in It, and 0 <H(x)<1,H(1) = 1. Moreover, H(x) satisfies the functional equation H(x) = S (G(x)) H (G(x)). If the source function S satisfies condition (S*), then H(x) is not identically one on It, indeed, for every x $ 1 in It, H(x) < 1.

3. If G is supercritical: x^° ¢ 1, then Hk (O) = H S (Gi(x°))= (S(x°))k 0, SO H(xO) = 0. Moreover, it is easy to see that H(x) _ 0 for all x 5 1 in It. For limGk(±) = x°(rx z 1) and hence S (Gk()) is bounded from 1. Thus only the subcritical case is of interest.

Theorem. If G is below-critical, and S is a polynomial of degree s, then H(x) is continuous.

Proof. We have H_-(x) - H,(x) = S (G(x))...S (G'-l(x)) (1 -S (G .(x))) 1-S (G^h (x)) < 1-S(G"(0))

Now, by Taylor's form, g2(0) > 1 + Ej m(n (0 - 1) = 1- yj m(n) > 1 - Vr'. Since r < 1, the latter is positive for all n sufficiently large. Now 1-S (Gn(0)) < 1-Es (i,...,jt) (1-Vrn(j) < 1-S(jl, ...,jt) (1 - Vrn) = 1-(1 - Vrn)s < 1- (1 - sVrn)<sVr^n. (See Appendix B). It follows that the sequence Hn(x) is uniformly convergent to H(x) on It, and hence the latter is continuous on this range (see Appendix A).

4. Since H,(x) = S(G)...S(G") = Hn_S (Gn), we have OHn/Oxj = OHn-_i/xj •S (Gⁿ ) + H_n-l (S (G)) /Oxj and \OHn_ /Oxj- OHn/G3xj < OHn-1/Oxj . (1 - S(Gn)) + H,O_l (S (Gn)) /Oxj. Now aHn_l/0xj <OS(G)/0x ₃+. . .+S(Gn-¹) /O ^xj and OS (Gn) /Ox^j = E (S/x)₀ (Og/x) (OSS/)Oi) (n)/ < E (S/xi) m⁾ < TE TWrn.

Thus the above absolute value does not exceed TW (r + ...+ rn^-~ ) sVrn +TWr' <Krⁿ(1+r +... + r-¹ ) < Krn/(1 - r). Hence by Appendix A, the n-sequences OHn/Oxj are uniformly convergent on It, the partials OH/Oxj exist, and lim OHn/0xj= OH/Oxj in It.

But (OHn/Oxj)₁ is the expectation of particles of type j in generation n, and the limit approached by this expectation is (OH/Oxj)₁.

Since we know H satisfies the functional equation H = S(G)H(G), we have OH/Oxk = EOS/Oxj dgj/OxkH(G) + S(G) ZOH/Oxj Ogj/Oxk. Setting x = 1, we obtain s (mjk -^S jk) (OH/O9xj), = - E (0S/0xj)I mjk, k= I,...,t.

Since G is below-critical, the determinant of the system is not zero and the expectation limits are uniquely determined. Thus follows the

Theorem. If G is below critical and the source function S is a polynomial, the limit function H possesses first partials on It, and the limit of the expectation of particles of type j in population n is the value at x = 1 of OH/Oxj. Moreover the latter limits are uniquely determined by the linear system E(mi ^k - bjk) (OH/0xj)l= -mjk (OS/OXj)l with non-vanishing determinant IM - II.

V—
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.

VI—
Total Progeny in Subcritical System with Source

1. Consider again the process described in IV. We found that the probability distribution in the n-th generation had generating function Hn(x) = S(G)S (G² ) . . . S (Gn) .

Here S (G') gives the distribution for the isolated system produced by the initial action of the source, S (Gn^-1 ) is that for the component due to the second action of the source, and so on. Regard the isolated component produced by the (n - k + l)st action of the source, with generating function at the n-th level (of the whole system) S (Gk). By an argument exactly analogous to that of V 1, one sees that the generating function for total progeny at the n-th level for this isolated component of the system is given by S (xic (x),... ,xtCz(x)).

It follows from the elementary laws of probability that the generating function Un(x)- un (il,j,jt) x .. .Xis Un(z) = S (xic(z)..., xtCt(x)). Here n (jl,...,jt) denotes the probability that, at the n-th level of the entire system, there should be a total of ji particles of type 1,..., jt particles of type t produced altogether, counting particles contributed by the source as well as all progeny of such particles.

2. Since Un(x) = Un- ₁(x)S (xici(x)), and for arbitrary x in It with all components less than one, the latter factor is less than or equal to S(x) which in turn is less than one (supposing condition (S*)), it is evident that lim Un(x) = 0 for all such x. Since however Un(1) = 1 for all n, Un(x) converges to a discontinuous limit function on It. Moreover, it is manifest that the expectations approach infinity so we cannot expect a simple theory.

Appendix A. We collect here some standard results from classical analysis.

A sequence of functions Fn(x) on It is said to be uniformly convergent on It in case (1) limFn(x) = F(x) exists for each x of It, and (2) for every e > 0 exists N so that n > N implies IF,(x) - F(x)l < e, all x in It.

Theorem. If all Fn(x) are continuous on It, and the sequence is uniformly convergent there, then the limit function F(x) is continuous on It.

Theorem. If Kn(z) is a uniformly convergent sequence of continuous functions on It, with limit function K(x), then the sequence of functions In(X) =- 0 Kn (Zl,X2, ...,xt)dzl is uniformly convergent to jXK(zl,x ₂,...,xt) dzl, and similarly for the other variables.

Theorem. If (1) Fn(x) converges pointwise on It to the limit function F(x), (2) the partials aF,n/xl exist and are continuous on It, (3) the sequence QFn/Oxl converges uniformly on It, then OF/Oxi exists on It and is equal to the limit of the sequence aFn/9Xl. Similar statements hold for the other variables.

Theorem.Fn(x) is uniformly convergent on It if and only if e > 0 implies existence of N such that for all n>N and all positive integers p, Fn+p (x) - Fn(x) < e on It.

Theorem. If F (x) is a sequence of functions defined on It, E Mn is a convergent series of non-negative numbers, and for all sufficiently large n, Fn(x) - Fn+l(x)\<Mn, n It then Fn(x) is uniformly convergent on It.

Appendix B Theorem. If 1 + h> 0 and n is a positive integer, then (1 + h)ⁿ> 1 + nh. Proof trivial by induction on n.

VII—
The "Time" Particle

1. It is of interest to note that, in the system with source, it is possible to regard the n-th population as essentially the n-th generation of progeny of a simple system of t + 1 types of particles produced from one particle of new type t + 1. Suppose that we associate x₁ ,. .., xt as before with the t original types of actual particles, and introduce a new type of particle with variable xt+l. Consider then the transformation V(x) of It+l, defined by the component functions vi(X) = gi(x) vt(x) = gt(x) X: _ (X1, .. ,Xt,Xt+i) Vt+l(x.) = Xt+lS(G(x)), x = (Xl,...,Xt,).

One verifies easily that the (t + 1)st component of the k-th iterate Vk(x) of V(x) is = xt+lS(G(x))...S (G(x)). Hence vk+1, the result of a simple iteration satisfies the relation t+l(Xl ,... ,Xt,Xt+l) = Xt+Hk(X₁, . ..,Xt).

The transformation V(x) fails to satisfy the restrictions we have imposed throughout on our generating transformations, for example vli/Oxt+l vanishes identically on It+l, and we have treated the problem independently for this reason. Nevertheless we shall be able to make use of the indicated simplification to the iterative case in a future report on the space of histories of a multiplicative system.

VIII—
Total Progeny as an Iterative Problem

1. In a similar way, the transformation Ck(x) of V may be produced by an iterative process. Let x₁ ,...,xt, be the usual variables associated with the t types of actual particles, and z,... ,Zt, be variables for a set of t types of dummy particles. Suppose probabilities of transmutation among the 2t types are defined by the following generating transformation L(x,z) of I2t, L1(x, z) = zig,(X,..,Xt) Lt(x, z) = Ztgt (X,...,t) Lt+l(x, z) = zi L2t(x,z) =- t . = (l,...,t) =(Z,...,Zt)

where the gi(x) are the components of the usual G(x). One verifies easily that ththe -th component (for i = 1,.. , t) L(x, z) of the k-th iterate of the transformation L(x, z) satisfies the relation L (x, x) = XiCk (Z)

If one examines the nature of the process induced by L(x, z) one finds that each time an actual particle of type xi is produced in a generation k, it is forced to produce in generation k+l a dummy particle of type Zi, as well as its actual progeny. Moreover, every dummy particle of type zi is forced to just reproduce itself in one for one fashion. Thus the total actual progeny is tallied by means of the one for one reproduction of the dummies through all generations. Thus if one sets zi = xi in generation k one totals the entire progeny including the actual particles produced from actual particles in the (k- 1)st generation and the dummies which total the whole previous progeny of actual particles. The extra xi factor is due to counting the zero-generation particle in the L process.