spectral representation of branching processes. ii

Z. Wahrseheinlichkeitstheorie verw. Geb. 5, 34--54 (1966)

Spectral Representation of Branching Processes. II* Case of Continuous Spectrum

SAMV~L KA~Ln~ and JAMES M c G ~ o o R

Received September 28, 1965

w 1. Introduction

This discussion is a n a t u r a l sequel to the preceding paper . W e refer the reader to the in t roduc t ion of [3] which conta ins backg round ma te r i a l for the p rob lem under cons idera t ion here.

Consider a b ranch ing process induced b y a p robab i l i t y genera t ing funct ion c o

/ (s) ~ ~ a k 8 ~, ak ~ 0 (/c ~ 0, 1, . . . ) a n d / ( 1 ) ---- 1. W e assume th roughou t t h a t k=0

/(0) > 0. The b ranch ing process X ( t ) , t = 0, 1, 2 . . . . is a M a r k e r chain on the non-nega t ive integers whose t r ans i t ion probabi l i t i es

P~j -~ P { X ( t -~ 1) = i ] X ( t ) ---- i} are defined b y

c o

(1) ~ Pi~s~ = [ / (s)] i , i = 0, 1, 2 . . . . j=0

Our a im is to inves t iga te the spect ra l p roper t ies of the m a t r i x P = [] PiJ[li.j=0. I f ]n (s) ~ ]n-1 ( /(s)) denote the i t e ra tes of ](8) then i t is qui te famil iar t h a t

the n s tep t r ans i t ion p robab i l i t y m a t r i x possesses the genera t ing funct ion

(2) ~ P58 J = [/. (8)]~. j=O

I n [3] we t r e a t ed the case where m = ] ' (1) > 1 or m ~ 1. I n the case m < 1 we requi red t h a t ] (s) be ana ly t ic a t 1. I t t u r n e d ou t unde r mi ld r egu la r i ty condi t ions t h a t P and i ts i t e ra tes a d m i t a spec t ra l decompos i t ion of the form

o o

(3) P ~ ~- ~ c nr 0I (r) y~l (r) i, ~ ~- O, 1, 2 . . . . r=0 n = 0 , 1 , 2 . . . .

where c ~ [ ' (q) and q is the smal les t posi t ive solut ion of [ (s) = s. B y in t roduc ing an app rop r i a t e H i l b e r t space we also p roved t h a t P acts as a comple te ly cont inuous t rans format ion .

W h e n m = / ' ( 1 ) = 1, P ceases to have eigenvectors and in fact P has only "cont inuous spec t rum" Throughou t , the r ema inde r of th is p a p e r i t is assumed t h a t m = ]' (1) = 1 and ](8) is ana ly t i c in a circle of rad ius 1 + s, e > 0.

* Research supported in part by Contracts ONR 225(28) and NI t t USPHS 10452 at Stanford University.

Spectral Representation of Branching Processes. I I 35

(7)

where

The prime objective of this paper is to establish, for the case m ~ / ' (1) = 1, a spectral representat ion of the form

co (4) p!~)~_ Se_n~Qj(~)dyj~(~) , i , j = 0, 1,2 . . . .

0 n = 1,2, . . .

where ~p~ is of bounded variat ion on [0, oo) and Qj (~) is a polynomial in the variable of degree ?', which vanishes at ~ ~ 0 ff ?" ~ 1.

We will prove the val idi ty of (4) subject to the following condition.

Condit ion I . Let g (s) be a probabil i ty generating function regular at s ~ 1. I f g (s) is not a constant then g' (1) > 0 and the inverse function g-1 (s) is then regular at s ~ 1 and there is a series expansion

(5) 1 - - g-1 (1 - - u) - - c~ u ~ k ~ l

with a positive radius of convergence. We say tha t g (s) satisfies condition I i /

(6) c~ => 0, k---- 1,2 . . . .

We will prove the validi ty of (4) under the assumption tha t / (s) satisfies condition I.

Not every probabil i ty generating function satisfies condition I. I n fact, for the generating function / (s) = 2e + (1 - - 3e) s + es 3 we find Cl > 0, c2 > 0 bu t c3 < 0 if s is small. Nevertheless, m a n y impor tan t generating functions do satisfy condition I. I t is easy to verify tha t / (s) = 1 - - a + as n (0 < a < 1), n ~ 1, 2, 3 . . . . and / (s) -=- (1 - - fl)~/(1 - - fls)v, 0 < fl < 1, y > 0, satisfy condition I. A large class of generating functions satisfying condition I, which includes the Poisson, the binomial, the negative binomial, and others, is described in Theorem 8, which is s tated here.

Theorem 8. Condit ion I is satisfied by every non-constant genera t ing /unc t ion o/ t h e / o r m

i = l l + ~ z g l--fli8 j = l

7>0, .4=>0, ]>~s>0, ~(~+fi~)<oo.

Theorem 9 of Section 5 describes another large class of generating functions which satisfy condition I. The two classes of functions overlap only slightly.

We remark tha t the class of all probabil i ty generating functions which satisfy condition I is closed under composition. That is, if g (s) and h (s) are generating functions which satisfy condition I, then so also is g (h (s)) and consequently all the iterates gn (s), n ~ 1, 2 . . . . satisfy condition I. A proof of this fact is given in Lemma 1. I t is no tewor thy tha t neither of the two classes of functions, described in Theorems 8 and 9, are closed under composition.

* We will mostly use the notation g-1 (x) to represent the inverse function of g, but sometimes we find it convenient to use the notation g-1 (x) as well.

3*

36 SAMUEL KARLIN and JAMES McGREGOR:

I t is shown in [2] t ha t the limit

(8) R(s) = lim { 1 1 / n-~oo 1 --fn(s) I --fn(O)

exists for Is ] < 1, uniformly for [s [ ~ 1 - - e, and

1 R(s) ~ 1 - - s as s - + l .

I t follows from (8) t ha t R (0) = 0, the power series expansion of R (s) about s = 0 has only non-negat ive coefficients, and tha t

R(/(s)) = R(s) + R(/(O)). I t is easily verified tha t

lim{ 1 1. } 1 n--+oo 1 - - f n + l ( O ) 1 - - fn (0 ) = ~ - / " ( 1 )

and hence 1 1

R(/(0)) = 2 - [ " (1 ) = K- > 0 .

The funct ion

(9) A (s) = K R (s)

plays a central role in the development of the representat ion (4). This function is regular in Is [ < 1, satisfies

(10) A( / ( s ) ) = A(s ) + l ,

(11) A ( s ) - ~ _ s as s - - ~ l , K - 1 = "(1)

and A (s) is a power series with non-negat ive coefficients, A (0) = 0. Since A' (s) > 0 for 0 < s < 1, and A (s) --> -~ co as s -+ 1, the inverse function

s = B (w) o fw = A (s) is regular in an open set containing the half line 0 < w < oo. I t satisfies 0 < B (w) < 1 for 0 < w < oo and

(12) / (B (w) ) = B ( w -4- 1).

F rom (12) follows [n(B(w)) = B ( w ~- n)

and now the subst i tut ion w = A (s) yields

/n (s) = B (n -~ A (s)).

Now P~) is the coefficient of sJ in the power series expansion of

(13) //n(S) = Bl (n -{- A (s)).

To obtain (4) we need to find the appropriate expansion for the r ight member of (13). To this end the following theorem is decisive.

Theorem 1. I / / ( s ) satisfies condition I then 1 - - B (w) is completely monotonic. The proof appears in Section 2.

Wi th the aid of this theorem we infer readily tha t oo

(14) B~ (w) = S e - ~ d~01 (~) 0


where Fi (~) is of bounded variation. Then (13) becomes

(15) [ /n (s)]~ = ~ e -n~ e-~(~)~d~ (~). 0

Now expanding both sides as power series in s and picking out the coefficient of sl yields the formula (4). We will rigorize the details of the above analysis in Sections 2 and 3.

In Section 4 we develop the spectral representation for a branching process with immigration. Section 5 is devoted to the classes of functions which satisfy condition I.

Probabilistic applications dealing with the strong ratio theorem, local limit theorems, and some application to some related genetic problems are deferred to other publications. Uniqueness of the generalized stationary measure is treated in [4].

w 2. The Generating Function of the Stationary Measure

We assume throughout that / is analytic at 1, m = 1' (1) = 1, and condition I holds (see the Introduction).

Our aim is to establish the spectral representation. For this purpose Theorem 1 serves critically. The following auxiliary lemma is needed for the proof of Theorem 1.

Lemma 1. I / /(s) and g (s) are probability generating/unctions satis/ying condition I then h(s) = / ( g ( s ) ) also satisfies condition I. I n particular all the iterates /n (s), n = 1, 2 . . . . satis/y condition I.

Proo/. We have

1 - 1-1(1 - u) = ~ c~uZ c~ -> 0 , l = 1

o o

1 - - g l ( 1 - - v ) = ~ y k v k 7k>=O k = 1

and hence

1 - - h 1 ( 1 - - u ) = 1 - - g - l ( / - i ( 1 - - u ) ) = 1 - - g - l ( 1 - - v )

where v = 1 - - / - i ( 1 - - u) is small when u is small. Thus

1 - - h - l ( 1 - - u ) = 7 k l u~ k = l l

is a power series with non-negative coefficients and the lemma is proved. We are now prepared to prove Theorem 1 which we restate in greater detail.

Theorem 1. Let /(w) satis/y condition I. Then 1 -- B (w) is completely monotone on 0 ~ w < c~, i.e., there exists an increasing /unction 0 (~) defined on [0, oo) o/ total variation 1 such that

r

( 1 6 ) 1 - - B ( w ) = ~e-w~dO(~) f o r a l l w ~ 0 . 0

Proo/. W e e x a m i n e

1 1 ( 1 7 ) 1 -- .f . (s) qn - - Rn(s)

38 SAMVEL K~RLI~ and JA~S ~IcG~Eoo~:

where qn = 1 - / n ( O ) . Since gn(s) converges to A (s) uniformly on compact subsets of ] s I ~ 1. The inverse funct ion Rnl(w) of Rn (w), tends to B (w) uniformly on bounded subintervals of [0, oo).

Solving Rn (s) ---- w for [n (s) gives

1 /n (s) ~- 1 w + q;1

and so

1 w + q~

Now, on account of L e m m a 1, we have o o

w 1 '~ (18) 1 R~l(w) = 1 [;1 1 + 1

where c(k n) ~" 0 and the convergence holds for all w >= 0. Each term 1/(w + 1/qn) k is tr ivially completely monotonic. Indeed, we have

the explicit representat ion cx~

1 1 Se_W~e_~/q=~k_ld~.

Since the coefficients in (18) arc non-negat ive it follows t h a t the sum 1 - - R~ 1 (w) is completely monotone. Thus

~xJ

1 - R ; 1 (w) = J" e-We 0n (~) d~ , w > 0 0

and On(~) >= 0 for all ~ >_ 0. Notice tha t On(~ e) has total integral 1 because R~I (0 )= 0. Bu t R~ 1 (w) tends to B(w) for w > 0. Therefore 1 - - B(w) is completely monotone and

1 - - B (w) = f e-We dO (~) ~ (w) >= 0 0

where 0(~) is an increasing funct ion on [0, ~ ) . Since B(0) = 0, we deduce by Abel 's t~heorem t h a t 0 (e) is a distr ibution function.

We can sharpen the result of Theorem 1 ff we assume slight;ly more on /(s). Thus

Corollary 1. L e t / ( s ) satis/y Condition I and also assume tha t / ' (0) > O. Then (16) holds [or ~ ( w ) > -- s / o r some positive s.

Proo[. Differentiating A ( / ( s ) ) = A ( s ) - ~ 1 and sett ing s ~ 0 leads to A'(0) / ' (0) A ' ( [ (0)) > 0. Therefore, A (s) is schlicht in a neighborhood of the origin and

consequently B (w) is analyt ic at the origin. The PRrNGSHEIM theorem (see WIDD]~X [5, page 58]) applied to Laplace t ransforms implies t ha t the half plane of convergence is as indicated in the Corollary.

I t is worth making a comment concerning Corollary 1. I f ] ' (0) ~ 0 then B (w) has a singulari ty at w ~ 0. The half plane of convergence in (16) is exact ly

(w) ~ 0 and no more. A specific example of this si tuation is furnished by /(s) ~ ao -~- (1 - - a0) s ~ (k ~ 2).

We now consider some properties of 0 (~).


L e m m a 2. The monotonic/unction 0 (~) occurring in (16) satisfies the asymptotic relation l im 0(~)/~ ~- K. In particular dO (~) has positive measure in every neigh-

~$o borhood o[ zero.

Pro@ I t is i nd ica t ed in (11) t h a t A (s) obeys the a sympto t i c law

l im (1 - - s) A (s) ---- K . s t l

Passing to the inverse funct ions s ~ B (w), th is l imi t formula says.

[1--B(w)]w--->K as w-->co or equ iva len t ly

oo

(19) ]e-w~dO(~),.~K/w as w - ~ c o . 0

W e will now use crucial ly the fact t h a t 0 (~) is an increasing funct ion. Indeed , because 0 (~) is increasing we can a p p l y a classical Tauber i an theorem (see [5], page 197) to (19) which yields the conclusion

(20) O(8)~K~ as ~ - + 0

as was to be shown. W i t h the a id of L e m m a 2 we de te rmine the ra te of decay of der iva t ives B(r)(w)

of B (w) as w -+ oo. Consider oo oo

(__ 1)r+l B(r)(w) : f e-we ~r dO (~) : f e -wr da (~) (21)

where 0 0

8 ,~(~) = S v r d O ( ~ ) .

0

F r o m (20) we deduce, b y in tegra t ion by par ts , t h a t

(22) a(u) = .fSrd0(8) -~ K ur+l o r -~ i as u ~ 0 .

Invok ing a welt known Abel ian theorem for Laplac8 t ransforms (see [5], page 181), we conclude t h a t

Kr! (23) ( - - 1)r+lBr(w) "~ Wr+l as w- ->co .

The above analysis demons t ra tes

L e m m a 3. When [ satisfies condition I the/unction B (w) obeys the asymptotic relations

K 1 - - B ( w ) ~ as w - - > c o

W

(24) Kr! ( - - 1) r+l B (r) (w) ~ ~ - as w -+ co, r = 1, 2 . . . .

The nex t two lemmas give some in format ion concerning the na tu re of 0 (~) for ~: large.

L e m m a 4. I / / ( s ) satisfies condition I and o o

(25) SdO(~) = 0 for some b > 0 b

40 SAMVEL KAXLII~ and JAI~ES McGREooI~:

then ](s) is regular on - - co < s < 1 , ] ' ( s ) > O o n - - co < s < 1 , / ( s ) - ~ - - co as

s -+ - - co a n d / ( s ) / s is bounded as s --~ - - c o . . F i n a l l y / ( s ) has exactly one zero on - - c o < s < l .

Proo/. T h e r e p r e s e n t a t i o n (16) b e c o m e s

b B (w) = 1 - - f e - u s dO (~) !r e w > O,

0

which shows t h a t B ( w ) e x t e n d s to an en t i r e func t ion , B ' ( w ) ) 0 for - - co < w < c o

a n d B ( w ) -+ - - oo as w --> - - co. I t fo l lows t h a t A (s) can be c o n t i n u e d a n a l y t i c a l l y

on t h e en t i r e h a l f l ine - - co < s < 1, A ' (s) > 0 for al l such s a n d A (s) -+ - - co as

s - ~ - - co. T h e f u n c t i o n a l e q u a t i o n

/ (s) ~- B(1 + A(s ) )

t h e n p r o v i d e s t h e a n a l y t i c c o n t i n u a t i o n o f / ( s ) on - - oo < s < 1 a n d shows t h a t

/ ' (s) > 0 a n d / (s) --> - - oo as s --> - - oo. S ince B (w) < B (1 ~- w) < 0 for w < - - 1

w e h a v e

f (B (w) ) B(1 + w) O < B ( w ~ - - B ( w ) < 1 , w < - - I

a n d hence / (s ) / s is b o u n d e d as s --~ - - co.

F i n a l l y , / ( B ( - - 1)) ---- B(O) ---- 0 s o / ( s ) has a zero on - - co < s < 1. T h e r e can

be no o t h e r zero s i n c e / ' (s) > O.

L e m m a 5. L e t / ( s ) sat is /y condit ion I and any one o/ the /ollowing:

(a) /(s) is regular and bounded on - - oo < s < 1

(b) / (s) is not regular everywhere on - - c o < s < 1

(c) / (s) is regular on - - co < s < I b u t / ' (s) is not > 0 everywhere on - - co < s < 1

(d) / (s) is regular on - - co < s < 1 but / ( s ) / s is not bounded as s -+ - - co.

(e) /(s) is a polynomial .

T h e n oo

(26) f d O ( ~ ) > O for al l b > 0 . b

Proo/. W e a s s u m e (26) is false so t h a t (25) holds. E a c h of t h e h y p o t h e s e s (a),

(b), (e), (d) d e a r l y v io l a t e s a conc lus ion o f L e m m a 4. Th i s c o n t r a d i c t i o n p r o v e s

t h e r e su l t for cases (a), (b), (c), (d). I n case (e) t h e h y p o t h e s e s / (0) > 0 , / ' (1) = 1

i m p l y t h a t t h e degree o f t h e p o l y n o m i a l is N > 2, so / ( s ) / s is n o t b o u n d e d as

s -+ - - co a n d we o b t a i n t h e conc lus ion as in case (d).

T h e o r e m 2. Let /(s) be a non-constant probabili ty generating ]unction o/ the type

(7) and m -~ / ' (1) ~ 1. T h e n / ( s ) satisfies condition I, and in the representation

oo

1 - - B (w) = .{ e-W~ dO (~), Ir e w > O, o

we have f dO (~) > 0 / o r every b > O. b

Proo/ , T h e m a i n b u r d e n o f p r o o f consis ts in showing t h a t / (s) satisfies c o n d i t i o n

I . Th is is t h e c o n t e n t o f T h e o r e m 8, wh ich we a s s u m e for t h e p resen t .


W e a s s u m e ] d O ( ~ ) = 0 for some b > 0 a n d de r ive a con t r ad i c t i on . Th is b

a s s u m p t i o n impl ies ( L e m m a 4) t h a t / (s) has e x a c t l y one zero on - - co < s < 1,

hence e x a c t l y one :q is d i f fe ren t f r o m zero a n d

1 + ~s ]~- 1 - - f l j /(s) = ev(s-1) ~ + c~ l i 1 - - fijs "

NOW since / (s) -+ - - e~ as s --> - - c~ ( L e m m a 4) we in fe r t h a t ? ~ 0 a n d / ~ i = 0

for all ]. T h u s 1 § ~ 8 / ( s ) - l + ~

which c o n t r a d i c t s t h e c o n d i t i o n m ----/ ' (1) = 1.

T h e o r e m 3. Let /(s) : 1 -k (s - - 1) g(s) where g(s) is a probability generating /unct ion o/ type (7) such that

Then /(s) is a probability generating /unct ion with / ' ( 1 ) - - - - 1 and /(s) satisfies condition I. I n the representation

oo

1 - B ( w ) = ~e-w~dO(~) , R e w >= O, 0

we have oo

f d O ( ~ ) > O for all b > 0 . b

Proo/. T h a t / (s) satisfies cond i t i on I will be p r o v e d as a coro l la ry of T h e o r e m 9.

W e a s sume t h a t f dO (~) = 0 for some b > 0 a n d o b t a i n a con t r ad i c t i on . b

B y L e m m a 4, /(s) -+ - - c~, as s --> - - cr so sg(s) -+ - - oo as s -+ - - oo. Also

/ ' ( s ) > 0 for - - oo < s < 1, so

g(s) + ( s - - 1)g ' ( s ) > 0 o r

f(8) 1 (27) g(s) ~ 1 - - s ' - - oo < s < 1.

B u t

f (s) ~ fit q(8) - ~ + ~ ~ + ~ ; 8 + ~ 1 =~j8

F r o m (27) we see g'/g < 1 for s < 0 wh ich impl ies t h a t all gi a re zero. Also f r o m (27), g'/g --> 0 as s -+ - - c~ so ? --~ 0. H e n c e

g(s) = ~ :-?j8 ' ~ j �9 0, ] = ]

a n d th i s con t r ad i c t s t h e f ac t t h a t sg (s) --+ - - oo as s --~ - - oo.

w 3. Spect ra l R e p r e s e n t a t i o n of PI~ ~ in the Case m = f ' ( l ) = 1

W e a s sume t h r o u g h o u t th i s sec t ion t h a t / (s) satisfies Cond i t i on I . Our s t a r t i n g p o i n t is t h e r e l a t ion (13)

(28) [ /n(s)V ~ B~(n + A (s)).

42 SAMVEL KA~LI~ and JAMES McGREGOm

B y Theorem 1 we know tha t the funct ion B (w) possesses a representat ion c ~ o o

(29) B(w) ---- 1 -- fe-w~dO(~)= Se-w~dVl(~) o o

where F1 is a signed measure of bounded variat ion arising from 0 and a ~ measure at the origin as follows; 0 is a finite positive measure on (0, c~) and ~Vl = 3 - - 0. I t follows f rom the convolut ion theorem for Laplace t ransforms tha t

o o

(30) Bi (w) = ] e-w~ d~oi (~) 0

where ~v~: (~) is the /-fold convolut ion of ~Ol and is a signed measure of bounded variation.

Combining (28) and (30) gives o o

(31) [/n (s)] l -~ ~ e-~%-A(~)~dYJi (~). 0

Now differentiate both sides ] times with respect to s and let s = 0. This is a familiar legitimate operat ion on Laplace transforms. The result is the formula

o o

(32) P~i = .[ e-n~ QJ (~) dy~i (~) 0

and Q1 (}) is a polynomial of exact degree ]. We have proved

Theorem 4. I / / ( s ) satisfies Condition I, / ' (1) = 1 and P = I[ Pi][]i,~ = o denotes the associated transition probability matrix o/ a branching process induced by /(s) then P admits the spectral representation (32) where Qj (~) is a polynomial o/degree ] and ~oi (~) is a signed measure o/bounded variation.

Corollary 2. Let / (s) satis/y Condition I and assume that/' (0) > 0 (see Corollary 1 o[ Theorem 1). The probability 1 - - / n (0) o I non-absorption by time n (n ---- l, 2, ...) where the initial population size is 1, determines a Hausdor# moment sequence.

Pro@ Consulting (28) and (29), we find t h a t o o

1 - - In(O) = f e -n~dO(~) n = O , 1 ,2 . . . . 0

where 0 (~) is a distr ibution funct ion (its total variat ion is 1) with no mass at the origin. Execut ing the charge of variables x = e-~ and da (x) = dO (-- log x) yields the display 1

1 - - In (0) = ~ x n d(l ( x ) , n = 0, 1 . . . . 0

which visibly affirms the fact t ha t {1 In (0)}~ = 0 s a Hausdorff moment sequence.

w 4. Spectral Representation for Branching Processes with Immigrat ion

I n this section we investigate the spectral representat ion for the transit ion probabi l i ty matr ix of a branching process with immigration. The transit ion probabil i ty matr ix is determined implicitly by its generating function

(33) ~ PijsJ = [/(s)]~ g(s), i = 0, 1 . . . . j = 0

Spectral Representation of Branching Processes. II 43

where /(s) and g (s) are probability generating functions./(s) can be interpreted in the traditional way as the probability generating function of the number of progeny per individual per generation. Individuals act independently and so the fluctuations of the population size in one generation due to a current population of size i is expressed in (33) by the factor [[(s)] I. The number of new individuals immigrating into the system in each generation is governed by the probability generating function g (s). The contributions arising from birth and death of the present population and new immigration are assumed to be independent.

A simple induction applied to (33) shows that the n step transition probability matrix has a generating function

(34) ~ pn. M = [ln (s)]~ g (]rt-1 (8)) ~ (ln-2 (8)) . .. g (8) ~2 ] = 0

We assume throughout this section unless stated explicitly otherwise, that /(s) and 9(s) are analytic at s = 1, /'(1) = m = 1 and g(s) ~= 0 for Is I ~ 1 (see Remark 4.1 below). The next theorem, of independent interest, is needed in estab- lishing the spectral representation for p n = I[ P~ rl.

Theorem 5. Let b = g'(1) and a = /"(1)/2 then

n

(35) lim n~/a~-~ g ( / ~ ( s ) ) = ~ ( s ) , ]s I < 1 , s 4 1 n--> oo k = 0

exists, z (s) is a~utlytic and ~: 0 [or all Is] <= 1, s =~ 1. The convergence is unqorm away / rom s = 1.

Proo/. Consider the sequence of functions

1 b log n Hn(8) = ~, ogg(/~(s)) + a k = 0

log n. b = logg(s) -? log[1 -- (1 -- g(/k(s)))] + a k = ]

The function Hn (s) is well defined since g (s) by hypothesis is never zero in Is [ < 1. Now recall the fact that /n (s) converges uniformly to 1 for Is I ~ 1. Consequently, it is permissible to expand the logarithm, yielding

"n (8 ) = - - ~ ~ =l-=(1--g(/k(8))rr -F-~ ]ogn. k = 0 r = l

I t is convenient to separate the terms r = 1. With this done we have

k = O k = O , r = 2

Now the Taylor expansion oo

g ( 8 ) = l + ~ b ~ ( s - - 1 ) ~, b l = b k = l

is valid in a neighborhood of 1 by assumption. Inserting this series in (36) and

44 SAMUEL KARLIX and J~ns McG~EooR:

performing obvious rearrangements produces the formula

Hn(s ) = ~b1(J~(s) - - 1)l + logn - - l(]~(s) -- 1) 1 k = 0 l ~ l k = 0 r = 2 1

�9 t co

= b ~ ( [ k ( s ) - - l ) + b l o g n + 5 5 b t ( / k ( s ) - - I )Z-- k = 0 k = 0 I = 2

k = 0 r = . \ I = 1

We will now prove the existence of the hmits, as n --~ ~ , of

(37) ~ (/~(s) - - l) + logn , a

k = O

$r oo

(38) ~ ~ b 1 ( f ~ ( s ) - - 1) l , k = 0 l = 2

and

k = 0 r = 2 l

where, the convergence is uniform on the region Is ] < 1, Is - - 11 ~ s > 0. For this purpose, we use the asymptot ic formula;

1 (40) 1 - - /n (S) = 1

- - + na + O(logn) ] - - 8

established in [2], page 20. The 0( ') te rm is uniform with respect to Is[ < 1, I s - - 1 ] ~ , > O .

By vir tue of (40), we deduce tha t

(41) ~ 1 - / ~ ( s ) - ;~ k = l

converges for Is[ G 1, s * 1. Indeed,

1

k = l k=l T-~+lca+O(logk)

E' 1" k=l a . k ~ - s + k a + O ( l o g n )

The terms of this sum when Isl < 1 and s 4 :1 are dominated by C(s)~z_.., logk = ]c2

k = l

where C (s) is uniformly bounded in the region Is I =< 1, ]s - - 11 ~ e > 0. Thus the assertion of (41) is vail& I t now follows tha t

[/~(s) - - 1] + logn 1 logn - - k=l a ~ / ~ ( s ) - - l + + a k = l - k = l

converges as claimed.


The proof t h a t (38) and (39) each converge is easier relying on the es t imate (see (40))

11 - / n ( s ) [ < O(~) : n

where C(s) is uni formly bounded for ]s [ ~ 1, [s - - 1 ] ~ e > 0. These convergence propert ies in conjunction show tha t Hn (s) tends to a l imit

as n --> oo. Exponent ia t ing the series defined in Hn (s) we conclude t ha t n

nO/a~I g(/k(S)) converges to ~(s) k = O

which is non-zero for Is] ~ 1, s =~ 1. The proof of the Theorem is finished. R e m a r k 4.1. The a rguments above can be adap ted to the case where g(s)

m a y vanish in ]s [ ~ 1. S ince /k (s) approaches 1 uniformly for Is I ~ 1, it is clear t h a t only a finite number of the functions in

(42) g(s) , g(/(s)) . . . . . g(/~(s)) .. .

m a y have zeros. I f we avoid these zeros then the analysis of Theorem 5 carried over furnishes the conclusion t h a t

l i m n ~ g(/k(s)) = ~(s) Is[ <= 1, s 4 1 n--> oo k ~ 0

exists and is analyt ic for ] s ] < 1. I n this circumstance, however, the function z (s) vanishes a t a point if and only if one of the functions in (42) vanishes a t the same point. Examina t ion of the limit in (35) leads to

L e m m a 6. There exists a non-null ]unction ~ (s) analytic in Is] < 1 satis/ying

(43) ~ (/(s)) g (s) = z (s)

/or which ze(r) (0) >= O, r - - O, l, 2 . . . . Proo/. The funct ion constructed in Theorem 5 manifes t ly satisfies the desired

properties. We tu rn now to the essential t a sk of generalizing Theorem 4 to the si tuat ion

of branching processes with immigrat ion. Le t A (s) be determined b y (8) and as before let B (w) denote its inverse function.

As pointed out in (28) we can express the i t e r a t e s / n (s) in the form

(44) /n (s) = B (A (s) + n).

Now referring to (43) we readily deduce the ident i ty

~(s) ( 4 5 ) g ( / n - 1 (8) ) ~ / ( / n - 2 (8)) . . . g (8) = ~ ( f n ( S ) ) "

Therefore, in view of (44) and (45), (34) becomes

B~(A (s) + n) ~(s) (46) P~.jsi : [ / n ( s ) J l g ( / n _ l ( s ) ) g ( / n _ 2 ( s ) ) . . . g ( s ) : z ( B ( A ( s ) + n ) )

]=o

The spectral representa t ion is derived f rom (46), by imposing fur ther restrict ions on the na ture of / (s) and g (s).

46 SAMUEL KARLIlV and JA~ES MCGREGOR:

Hencefor th , we postula te the following: Assumpt ion A . (i) The probabi l i ty generat ing funct ion /(s) satisfies condition I (so Theorem 1

is applicable). (ii) The probabi l i ty generat ing funct ion g (s) is meromorphie of type (7).

The nex t series of l emmas culminates in the following theorem.

Theorem 6. Let /(s) satis/y Condition I. Let ~ (s) be defined as in (35). Then [~ (B (w))] -1 is completely monotonic.

We divide its p roof into easy stages.

L e m m a (i). The/ol lowing operations preserve the class o/ completely monotone (c. m.) /unct ions . (a) I f R(w) and S(w), (w ~ 0) are c .m. and ~ ~ 0, fi > 0, then ~ R ( w ) ~- f iS(w)

and R(w) S(w) are c .m. (b) I f Rm(w) --> R(w) for all w ~ 0 and Rn(w) is c .m. (n = 1, 2 . . . . ) then R(w)

is c .m. These assertions are famil iar and can also be rout inely proved. The reader m a y

consult WIDDE~ [5], for fur ther details.

L e m m a ( i i ) . I / 1 - - B ( w ) is c.m. and O < B(w) < 1, (O < w < oo), then 1 - - f iB(w) , 0 ~ fi < 1 is c.m.

Proo/. Observe t h a t

1 - fiB(w) = (1 - - f i) + f i [ 1 - - B ( w ) ) ] .

Now the posit ive cons tant 1 - fi is t r ivial ly c .m. and 1 - B(w) is c .m. by hypothesis . Applying L e m m a (i), pa r t a yields the desired conclusion.

L e m m a (iii). Under the conditions o / L e m m a (ii) the /unction [1 + :r -1, o: > 0, is c.m.

Pro@ Consider the representa t ion

I + ~ B ( w ) - - l @ ~ o: B(w)) - - l @ ~ k = 0 \ l + ~ ] - - l + : t ( 1 - -

L e m m a (i), pa r t a tells us t h a t each t e r m of index/~ ~ 1 is c .m. A cons tant is c .m. (this is the t e r m for Ic = 0). Again appeal ing to L e m m a (i) we conclude t h a t the sum is c .m.

L e m m a (iv). Under the conditions o / L e m m a (ii), the /unction e -€ (y ~ O) is c . m .

Pro@ The proof paraphrases the analysis of the preceding l e m m a using the ident i ty o o y~

e - Y B(w) __ e-V eT(1- B(w) ) _ e - ~ ~ - - _ ~ V . ( 1 - B ( w ) ) ~ .

k = 0

Combining l emmas (ii)--(iv) and invoking appropr ia te ly l emma (i), we infer

L e m m a (v). Let the conditions o/ Lemma (fi) prevail. Then

~I (1 - ~ B ( ~ ) )

(47) i = 1 e - r B(w)

f l (1 + ct~ B(w)) i = 1

is c.m. provided 0 <= fii < 1, 0 <= ~ (i = 1, . . . , n) and y >= 0 hold.

Spectral Representation of Branching Processes. II 47

We can let n -~ 0o in (47) again appealing to Lemma (i), part b provided both products converge. This limiting process is certainly justified if the parameters

c ~

obey the restrictions 7 ~ 0 , 0 ~ f i t ~ l , 0 ~ c c i , ~ ( ~ + f l ~ ) ~ c ~ . The limit i = l

function has the form

(48) K i = 1 e_YB(w )

(1 + ~ B(w)) i = 1

(K is any positive constant). Comparing (48) and (7) and recalling Assumption A (ii) we have demonstrated

Lemma (vi). Assume that 0 ~ B (w) ~ 1 /or w ~ 0 and that 1 -- B (w) is c.m. I / g (s) is a probability generating/unction o / t h e / o r m (7) then

1 (49) g(B(w)) is c.m.

Henceforth B (w) stands for the inverse function of A (s) defined in (8).

Lemma (vii). I / A s s u m p t i o n A holds and B (w) is defined as indicated above, then

1 is c.m. /c ~ 0, 1,2 . . . .

g(f~(B(w)))

Proo/. Recall t ha t / ~ (B (w)) ---- B (w +/c) . But

1 1

(50) g(f~(B(w))) - - g( B(w 4- k,))

involves merely a translation of the independent variable of (49) in the positive direction. I t trivially follows owing to Lemma (vi) and noting (50) tha t [g(/k(B(w)))] -1 is c.m.

We are in possession of the ingredients required to prove Theorem 6. Proo/o / Theorem 6. Lemmas (i) and (vii) in conjunction show tha t

1 is c.m.

/ t

n b/a I~ g ( f ~ (B (w ) ) ) k = o

Letting n -~ 0% we deduce tha t 1/[z(B(w))] is c.m. as was to be shown.

Corollary 3. Under the conditions o] Theorem 6, we have the representation o o

1 - S e-w~ d~ (~) ~(B(w)) o

where q~ (~) is an increasing/unction o/bounded variation. The fact tha t g (B (0)) is finite guarantees the property that ~ (~) is of bounded

variation.

Corollary 4. Under the hypothesis o/ Theorem 1, Bi (w)/7~(B (w) ) admits the representation

o o

B~(w) - ~ e-us d~t(~) (51) ~(B(w)) o

"where y)~ (~) is a signed measure o/ bounded variation.

48 SAMUEL KAOLIN and JAMES MoGI~OOR:

W e are now ready to tackle the p rob lem of the spectral representa t ion

Theorem 7. Let A s s u m p t i o n A hold. Then

o o

(52) P('~') = f e-n~ Q1 (~) &f f (#), i, j -~ O, 1, 2 , . . -~3 o n =- O, 1, 2 . . . .

where Qj (.) is a po lynomia l o/degree ] and tvi is a signed measure of bounded variation. Proo[. Referring to (46) we know t h a t

(53) f p~.iM = B*(A(s) + n) 7~(s). ~ (B(A(s ) + n)) j=0

B y vi r tue of (51) we can represent the r ight side of (53) in the form o o

(s) ~ e-~-A<~)~dw~ (~). o

Picldng out the coefficient of s] (see the a rgumen t af ter (31)) we get (52). I n order to prove t h a t Qi (~) is of precise degree ] we need mere ly observe t h a t

(0) > 0 in the ease a t hand when g (s) is of t ype (7). Actual ly, ~ (s) never vanishes for ]s] < 1 as noted in Theorem 5.

W e close this section with an example of the calculation of (35). Le t

t ( s ) = e + (1 - ~ )~ 1 - c s g (s) = eS-1.

The i tera tes o f / ( s ) are easily found, in fact

1 1 1--f(s)-- 1--* +ct' o:=c/(1--e)

and hence

1 1 i - f ~ ( s ) - - 1 - s

q- no~.

We introduce a new variable x "by se~ung" ~x = 1/(1 - - 8), and then

1 1- f . ( s ) - -~(x + n). (54)

Now

(55) n

n(~-~)~__]-[g (/~ (s))

= n1/~ e x p ~ [/~ (s) - 1] k = 0

n 1

Using the known formula, EgD~Su et al. [1], vol. I , p. 15,

lira l o g n o ~ - - /'(x) n - - > o o


we obtain f rom (55) r'( I-~I

1 - - c \ c ( 1 - - s ) ] g0 (S) = lira n(1-c)/cl~ g (/~ (s) ) = exp

n-->oo k = 0 c / , ( ~ ) 1 - - c "

w 5. Discussion of Condition I

Our ma in purpose in this section is to prove Theorem 8 which we res ta te as follows.

Theorem 8. Let / (s) be o/ the [orm (7) and not a constant, and let h(w) be the inverse/unction, so that

(56) h(/(s))----s near s = l .

Then

( - - 1 ) r -1 dr ] --~_0, r = 1 , 2 , - dwr -h (w) [ w = l . . . .

By repeated differentiation of (56) and evaluat ion a t s = 1 we obta in the relations

1 = h'(1)/'(1), 0 = h ' (1) /"(1) + h " ( 1 ) ( / ' ( l ) ) 2,

0 : h ' (1) / ' " (1) - } - h"(1)31'(1)/"(1) ~- h" ' (1 ) ( / ' (1 ) )a

and generally t

(57) O=~h(~)(1) ~ b ......... ( r , k ) • ( / ' ( 1 ) ) ~ 1 ( / " ( 1 ) ) ~ 2 . . . ( / ( r ) ( 1 ) ) ~', ~,*~] C{1,..,,~r

where the inner sum, the coefficient of h(~) (1), is extended over all selections of r integers ~i . . . . . ~r satisfying ~l ~ 0 . . . . . ~r ~ O,

(58) ~ ~i = k, ~ i~l = r i ~ l i = 1

and b . . . . . . . . . (r, k) are non-negat ive integers which do not depend on the func- t ion /. We will denote the coefficient of h(k)(1) by Jr, k ; / ] , and if there is no danger of confusion, s imply by [r, k]. Thus the general relat ion is

=~ _ 0 h(k) (1) Jr, k/ r ~-- 2, k = l

1 = h ' (1)1 ' (1 ) (r = 1 ) .

I t is easy to ver ify t h a t Jr, r / = (/ ' (1))r and [r, 1] ----/(r) (1). The equat ions can be solved to give h(r) (1) in te rms o f / ' (1 ) , . . . , / ( r ) (1). Cramer ' s rule produces the formula

(59) h(r) (1) = ( - - l) r-1

[2 ,1] , [2 ,2] , 0 o . . . 0 [3, 1], [3, 2], [3 ,3] , 0 . . . 0

[r, 1], [r, 2] . . . . . . . . . . . . . . [ r , r - - 1 ]

(]'(1))l+2+...+r

Z. Wahrscheinlichkeitstheorie verw. Geb., Bd. 5 4

50 SAMUEL KhRLIN and JAM~s McGREGOR:

The proof of Theorem 8 is therefore equivalent to showing tha t the determinants in (59) are all non-negative, tha t is

i [ 2 , 1 ] , [ 2 , 2 ] , 0 . . . 0 [3, 1], [3,2]

> 0 . (60) At( l ) = =

i [r,']], [r, 21 . . . . . . [ r , r - - 1] ]

We will first show tha t Air(/) ~ 0, r = 1, 2 . . . . provided ] is of the form (7) and

m = / ' ( 1 ) < 1. We recall f rom [3] t ha t

(61) lim ,fn(8) -- 1 mn - - A ( s ) , ]sl ~ l ~ - e n ---> o o

for some s ~ 0, and A(1) : 0, A'(1) --~ 1. Moreover A(s) satisfies A ( / ( s ) ) -~ m A (s). The funct ion B(w) inverse to w ~-- A(8) is regular in a neighborhood of the segment 0 ~ w ~ oo and

(62) / ( B ( w ) ) : B ( m w ) , B(O)~-- l , B ' ( O ) : I .

I t was established in [3, Theorem 4] t ha t

(63) ( - 1) r-1 B(r) (0) > O, r : 1, 2, 3 . . . .

Since A and B are inverse functions, and A'(1) ~ 1, it follows tha t

B(r)(O) = (-- 1)r-l Ar (A) , and hence

Ar(A) ~ O, r - ~ 1 , 2 . . . .

By differentiation of A (/(s) ) ~ m A (s) we find T

(64) mA(r) (1) : ~ A(~)(1)[r, k;]]. k ~ l

We consider the formula with / replaced by the function

f(ss) (65) g ( s ) - f ~

and A (8) replaced by the corresponding A e (s), where s is a positive parameter which will approach zero. Note t h a t g(1) = 1 and

me ~-- g'(1) = sl ' (s) / /(s)

which --~0 when s - > 0 because ](s)---> /(O) > 0. More generally s-rg(r)(1) /(r)(s)//(s) has a finite limit as s - > 0 , and hence, in view of (58), we find

lim s-r[r, k; g] = [r, k; / ]* / [ / (0 ) ] ~, where [r, k ; / ] * is the same as [r, k ; / ] bu t ~ - - > 0

with the derivatives of ] evaluated at s ---- 0 instead of 8 ---- 1. F r o m (64) in the

form r

(66) m~AT)(1) _ sf'(e) A~')(1) ---- ~A~k)(1)[r , k; g] f(s)

Spectral Representation of Branching Processes. I1 51

we can now deduce, b y induc t ion on r, t h a t sl-rA~r)(1) has a finite l imi t as s -> 0 for r ~ 1. B y using (66) again we can eva lua te the l imit . Since

e-(r-k+1) [r, ]c; g] -> 0 for ]c = 2, 2 . . . . . r

we find, in view o f A : ( 1 ) = 1,

f(r)(O) f '(0) l im sl-rA(~r)(1) = lira s-r/r, 1; g] - - f(0)

(67) f(0) , ~ o ~ 0

I t follows f rom (67) and (58) t h a t

l im sk-r [r, k; A , ] ---- [r, k ; / ]* / [ / ' (0 ) ] ~ . e-->0

W e a p p l y this to the known inequa l i ty A t ( A , ) > 0, and ob ta in

0 g lira s (1-~)+(2-r) +"" + (~- l - r ) Ar (A ~) ~-- A* (/) /[ / ' (0)] r(~-1)/2 e-~0

where A r~(/) is the same as A r( / ) bu t wi th the der iva t ives o f / e v a l u a t e d a t s ~ 0 ins tead of s = 1. Thus

(68) At*(/) ~ 0.

To ob ta in (60) f rom (68) we a p p l y (68) to the f u n c t i o n / * (s) ~ / (as ~- 1)//(a ~- l ) where a ~ 0 is small . Since / (s) is of the form (7) we have

1 - /~ %-1 ) whieh is again of the form (7) if a is posi t ive and small . Hence we can a p p l y (68),

(69) A*(/*)>O, r = l , 2 . . . .

F r o m (69), (58) and the iden t i t y r f(r)(1)

(/*)(r)(0) -~ a f ( a + l )

we deduce (60). Theorem 8 has now been proved for the case m ~ / ' ( 1 ) ~ 1. The resul t when

m ~ 1 follows by a t r iv ia l con t inu i ty a rgument . W e close wi th a br ief discussion of a second class of genera t ing funct ions which

sa t is fy condi t ion I . L e t / ( s ) be a p robab i l i t y genera t ing funct ion regular a t s ~ 1 w i t h / ' ( 1 ) ~ 1.

T h e n / ( s ) ~ 1 -~ q~(s -- 1) where ~(u) is regular a t u : 0, ~0(0) : 0, ~'(0) : 1. Le t u ~ ~p(v) be the inverse funct ion, i .e. , the solut ion of v ~ ~0(u) near u ~ 0. Then the asser t ion t h a t / satisfies condi t ion I is equiva len t to the asser t ion t h a t ~0(v) has a l t e rna t ing coefficients, (--1)r~p(r)(0) ~ 0 for r ~ l , 2 . . . . .

Theorem 9. I/u/q~ (u) has alternating coe/ficients then / satisfies condition I. F o r the proof we will need the following lemma.

Lemma. I / ~fl (x) and ~f2 (x) have power series with alternating coe/ficients then ~Pl (x) ~f~ (x) has alternating coe/ficients.

Proo/. If yJl(x) ---- ~ an xn and ~o2(x) = ~ bn xn then the n TM coefficient of

~1(x)~02 (x) is ~. ak bn-k. Using the hypothes is , we ge t k = 0

(-- 1 )n~a~bn-k = ( - - 1)kay( - l )n-~:bn-k ~ 0. k ~ 0 k ~ 0

4*

52 S~V]~L KA~LI~ and JAMES McGRECOR:

Proo/ o/ Theorem 9. Let

u = ,fl (v) ~- v -~ b2 v 2 q- b8 v a q- ""

By Cauehy integral formula we have

(70) b n - - 21if ~(v)dv v n ~ + Y - ' n ~ 1 F

where /" is a small contour about the origin. Under the subst i tut ion v = ~ (u), (70) passes into

Mow

d 1 du

1 / q~' (u) du 1"* bn-~- 2~-~iF * [~0~)_]n+l a n ' = ~ ) ( F ) ,

= coefficient u n-1 in ~'(u)

1

SO

The hypothesis combined with the Lemma asserts t ha t

1 [q~) ]n - - Co -~- Cl U -~- ' ' ' -~ C n - l U n - I -~ ' ' "

has coefficients al ternating in sign. Wi th reference to (71) we find tha t

b n = - - c n - l { 1 n - 1 } n - - Cn-:n ' n>l__

which clearly alternates in sign since c~ possess this property.

Corollary 5. I / /(s) ~ 1 + (s - - 1)/ l (s) where / l(s) is a probability generating /unction o/ type (7) with

(72) 0 < 5r + ~ ~i -4- ~ fij < 1 i j

then /(s) is a probability generating/unction which satisfies condition I . Proo/. We have

oo / (s) = 1 - - ao ~- ~ (an-1 -- an) ~n

n = l where

oo /1(8) = Z a n s ~ .

n=0 The left hand inequali ty (72) reduces to al/ao ~ 1 and so a0 - - al ~ 0. Since /l(s) is of type (7) the ratio an~an-1 is non-increasing and hence

a n - l - - a n ~ O, n = 1, 2 , . . . .

Thus /(s) is a probabil i ty generating function, and it is trivial to ver i fy / ' (1) = 1.


Finally in the notat ion of Theorem 9,

u 1 ~(u) - - A(1 + u)

which clearly has al ternating coefficients s ince/1 is of type (7). R e m a r k 5.1. Funct ions of type (7) rarely satisfy the hypotheses of Theorem 9,

so tha t the class of funct ion of Theorem 8 is almost disjoint from the class of functions of Theorem 9. For example none of the functions

1 / ( 8 ) = ( 1 - - p + p s ) S ' P - - N ' ~ V = 3 , 4 , 5 . . . .

/ ( s ) = e8-1 ,

/(s) = \ l _ p s / p - ~ + ] , ~=>2

satisfy the hypotheses of Theorem 9, but they are all of class (7). R e m a r k 5.2. There is an analog of Theorems 8 and 9 for the case where

/'(1) --~ m > 1 which we state as the following theorem.

Theorem 10. L e t / ( s ) be a probability generating/unction o/ type (7) /or which / ' ( ] ) --~ m > 1 ,and ](q) ~ q (0 < q ~ 1) or alternatively suppose / ( s ) is o/ /orra /(s) = q + (s - - q)/* (s) where /* (s) possesses the representation (7).

Let h*_l(w ) be the inverse/unction o / / ( s ) - - q = h(s - q) ~ h*(s) ~ w which is well defined in a neighborhood o/ w ~ O. Consider the power series expansion

c ~

(73) lwl <

Then

(74) ( - -1) ~ - l c k ~ 0 , k = 1 , 2 , 3 . . . .

The proof is identical to t ha t of Theorems 8 and 9. There is an impor tan t converse result which we now describe.

Theorem 11. Suppose / ' (1 ) ~ m > 1 a n d / ( q ) ~-- q (0 ~ q ~ 1) holds and de- fine the/unction h* (s) by the relation h* (s) = h (s - - q) = /(s) - - q. Assume that the expansion (73) exhibits coe/ficients o/ alternating signs as in (74) and let B (w) be the inverse/unct ion o / A (s) associated with the branching process induced by the p. g./. /(s) (see [3, section 2]). Then

( - - l ) r - l B r ( O ) ~ O , r - ~ 1 , 2 , 3 , . . . .

Proo/. We start with the relation

/ ( B ( w ) ) = B(cw) , w > O, c = / ' ( q )

or wha t is the same

(75) B(w) = / _ ~ ( B ( e w ) ) , w > O.

Notice tha t /(_r) a (q) ~-- h*_(~ ) (q), r ~ 1, 2, 3, . . . . Now successive differentiation of (75) and evaluation at w -~ 0 produces the formula (see (57))

(76) B(r) (0) ~- e r ~ h(k_ ) (q) ~. b ....... ~ (r, ]c) X k = l al,...,c~r

X (B'(0)) ~' (B"(0))~ .. . (B(r)(0)) ~"

where the ~'s are subject to the conditions as described in (58).

54 S. KXl~LI~ and J. MeG~EGO~: Spectral Representation of Branching Processes. I I

W e proceed b y induct ion. Suppose we have a l r e a dy p roved the inequal i t ies

(77) ( - - 1 ) k - l B ( ~ ) ( 0 ) ~ 0 , ] c : l , 2 , . . . , r - - 1 .

Separa t ing the t e r m / c : 1 f rom the first sum in (76) and not ing t h a t

h* [d _1 ( w ) ] / d w I w = q = ] /c

we have

(78) B(r)(O)(1--cr-1)=cr~h(~_)(q) ~ b ....... ~ ( r ; k ) X

• (B ' (0)) : ' (B"(0)) ~ . . . (B(r)(0))~"

where necessar i ly ~r = 0 in view of the res t r ic t ions

~ = / c and i ~ = r , ~ 2 . i = 1 i = 1

The induc t ion hypo thes i s tells us t h a t the sign of the inner sum is ( - - 1)z~,(i-1) h*(k) 1)k-1. = ( - - 1 ) r-~ and because of (74) we also know t h a t sign -1 (q) = ( - - I t

follows f rom (78) t h a t ( - - 1) r-1 B(r) (0) ~ 0 and the induc t ion s tep has been ad- vanced. The case r = 1 is t r iv ia l . This completes the proof.

R e m a r k 5.3. The above converse asser t ion implies a new resul t for the cases m > 1 and m < 1 t r e a t e d in [3]. Specif ical ly i t impl ies t h a t the spect ra l represen- t a t ions de r ived in Theorem 4 of [3] are va l id for n ~ 1 if l (s) satisfies condi t ion I in the ne ighborhood of the smal les t pos i t ive fixed po in t q = / ( q ) (0 < q ~ l ) as descr ibed in Theorem 10.

References

[1] ERD:~LYI, A., W. MAGNUS, F. OB~,RItETTINGER, and F. TRICOMI: Higher transcendental functions. New York: McGraw Hill 1953.

[2] HARRIS, T. E.: The theory of branching processes. Berlin-Gbttingen-Heidelberg: Springer 1963.

[3] K ~ L I ~ , S., and J. McGREGOR: Spectral theory of branching processes, I. Case of dis- crete spectrum. Z. Wahrscheinlichkeitstheorie verw. Geb. 5, 6--33 (1966).

[d] - - - - Uniqueness of stationary measures for branching processes and applications. To appear in Proc. Fifth Berkeley Sympos. math. Statist. Probab.

[5] WIDDER, D. V. : The Laplace transform. Princeton University Press 1941.

Department of Mathematics Stanford University Stanford, California 94305 U.S.A.

spectral representation of branching processes. ii

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