some results on multitype continuous time markov branching processes

7/29/2019 Some results on multitype continuous time Markov Branching processes

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Some Results on Multitype Continuous Time Markov Branching Processes

Author(s): Krishna Balasundaram AthreyaReviewed work(s):Source: The Annals of Mathematical Statistics, Vol. 39, No. 2 (Apr., 1968), pp. 347-357Published by: Institute of Mathematical StatisticsStable URL: http://www.jstor.org/stable/2239026 .

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The Annal8 of Mathematical Statistics1968,Vol. 89, No. 2, 347-357

SOME RESULTSON MULTITYPECONTINUOUSTIME MARKOVBRANCHINGPROCESSES1

BYKRISHNA ALASUNDARAMTHREYAStanfordUniversity

1. Introduction.Oflate therehasbeena lot of interest n multitypecontinuoustime Markovbranching rocesses(MCMBP).t wasrecentlynoted that there is afundamental and yet simple connection between classical urn schemes likePolya's and Friedman's,etc. and multitype continuous ime Markovbranchingprocesses see [1]).Problemson urnschemeshave theircounterpartsn branchingprocessesand it is while attemptingto solve these that the authorfelt the needfor a systematicstudy of the MCMBP.The presentpaper s a partialanswer othis need (see also [1], [2], [3]).In this paper we develop in a systematic way some basic properties of multi-type continuous imeMarkovbranchingprocess.Althougha fewof theseproper-ties areelementaryandnot entirelynewin contentwe present hem here forthesakeof completeness.But therearetwo veryimportantresultswhich we believearenew. We prove hem underminimalassumptions. See Section2 fornotationsandpreliminaries.) et {X(t); t _ O}be a MCMBPandlet A bethe infinitesimalgenerator f the meanmatrixsemigroup M(t); t > O}whereM(t) = ((mi1(t)))and mij(t) = ((E(Xj(t) IXr(O)= 83ri, = 1, 2, ... , k)))and 8ij areKroneckerdeltas. Assuming positive regularity and nonsingularity of the process we establishthe following:THEOREM. Needingnothingmorethantheexistenceof thefirst moments,wehave

limtoo X(t, w)eXlt - W(w)u exists wp 1where W( w) is a nonnegativenumerical valued random variable, Xi is the maxi-mal realeigenvalue f A and u is an appropriatelyormalized ector atisfyingu*A = Xiu* where u* is the transpose of u.

THEOREM4. LetXi > 0 so thatP{X(t) = 0 for some t} < 1for any nontrivialmakeupand assume second momentsexist. Thenlimt,- P1O < xl < W _ x2 < oo, (v.X(t) - eXltW)(v.X(t))-T1y}

= P{O < Xl < W < x2 < 00}((y/cf)wherev is an appropriatelynormalizedvector atisfying Av = X1v, an appropriateconstant nd 1i(x)is thenormaldistributionunction.Here is an outline of the rest of the paper. In Section 2 we describe our set upand construct some martingales. Section 3 establishes Theorem 1. Assuming the

Received 18 October 1967.1Research supported in part under contracts N0014-67-A-0112-0015 nd NIH USPHS10452 at Stanford University.347

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348 KRISHNA BALASUNDARAM ATHREYAexistence of second moments we study in Section 4 the growth behavior ofE It X(t) 12where is an eigenvectorof the matrixA. The last section developsarepresentationor X(t) - eXltWu nd uses it to prove Theorem4.This paper ormsa minorpart of- he author'sdoctoral hesis at the Departmentof Mathematics,Stanford.The author s much indebtedto his adviserProfessorS. Karlin orhelp andencouragement.

2. The set up, mean matrix and martingales.2.1. The setup. We start with a strongMarkov,continuous imek dimensional(2 < k < oo) branchingprocess {X(t); t > 0} definedon a probability pace(0, i, P). That is, {X(t, w); t > O} s a stochasticprocesson (Q, i, P) such that(i) The state spaceis the nonnegative nteger attice in k dimensions.(ii) It is a Markov chain with stationarytransitionprobabilities nd strongMarkov with respect to the family gt of a-algebras where gt = a{X(u, w); U < t}and a{D} standsfor the sub a-algebraof 5f generatedby the family D of realrandomvariableson (Q,5)(iii) The transitionprobabilitiesPi,j(t) satisfy

(ZEPi,j(t)s ) = Jk_l (Zj Pe,ij(t)si)irwhere i-(il i2 X ik), es = (a, Xi2 8ik)8zj= 1 if i = j

=0 if i s j.Let the associated nfinitesimalgenerating unctionsbe(1) ui(s) = ai[hi(s) - si] for i = 1, 2, .. , k,

whereO < ai < oo, s = (Si, S2, * , Sk), O ? si ? 1 and h(s) is a probabilitygenerating unction.We makethe followingbasicassumption.ASSUMPTION 1.

(2) Ohj(s)/OsjI8=(1,...j,) oo for all i and j.Under (2) it can be proved using either the theory of branchingMarkovprocessesdeveloped by Ikeda, Nagasawa and Watanabe[9] or the theory ofminimalprocessesas in Chung [5]that a branchingprocess{X(t); t > O}of theabovespecification xists. Also from the sametheoryone could take the samplepathsto be rightcontinuousn t wp 1.It is very suggestive o thinkof X(t) = (Xl(t), , Xk(t)) as the vectorde-notingthe sizes of the populationat time t in a systemwith k types of particleswhere(i) a tvpe i particlelives an exponential ength of time with meana-l' andon deathcreatesparticlesof alltypes accordingo a distributionwhosegeneratingfunction s given by hi(s),(ii) all particlesengender ndependent ines of descent.This property s thebasicfeatureof a branchingprocess.

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MULTITYPE CONTINUOUS TIME MARKOV BRANCHING PROCESSES 349A moment's reflection s enough to justify the followingrepresentationofX(t + u, co)as

(3a) X( t + u, co)- t=i ZX5w X (u, c)where{Xt'(u, w), u > 0} is the vectordenoting he line of descentfrom thejthparticleof type i living at time t. Without loss of generality (see Chapter6 in[8])onecan assumeQand 5Yo be "big"enough o makeXt'(u, c) a 5:-measurablefunctionand conditionally conditionedon X(t)) mutually independent.In termsof generatingunctions(3) becomes(3b) f(s, t + u) = f(f(s, u), t)where(3c) f(s, t) = (fi(s, t), f2(s, t), ..., fk(s, t))and

fj(s, t) = E(siX1(t)S2X2(t) ... SkX(f) IX(O) = ei)Onerelates the ui(s) to f(s, t) as follows:The Kolmogorov orwardand back-ward differential quations orPi j(t) lead to the followingn terms off(s, t).(3d) (forward) ofi(s, t)/Ot = Ej=, (afi(s, t)/asj)uj(s), i = 1, 2, ky.(3e) (backward) afi(s, t)/Ot = ui(f(s, t)), i = 1, 2, kthe initial conditions for both the systems being fi(s, 0) = si, i = 1, 2, k, k.2.2. Mean matrix.From(2) it follows (see Chapter5 in [8]) that(4) mij(t) = E(Xj(t) IX(O) = ei)is finite for allO _ t < oo, andj.If we set(5) M(t) = ((mij(t)))kXk, for t > 0we get, using (3b), the semigroupproperty(6) M(t + u) = M(t)M(u) for all t, u > 0.We alsohave using (3d) or (3e) the continuitycondition(7) limt,oM(t) = I.

Let(8) bij = chi(s)/dsjfs=(l,l,... ) -bij,

aij= aibij A = ((aij))kxk for 1 < i, j ? k.Thenit is well knownthat M(t) has the representation(9) M(t)=el for t 2 0.



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350 KRISHNABALASUJNDARAMTHREYAWe next make the basic assumptionof positiveregularity,viz., that thereexiststo > 0 andfinite suchthat(10) mij(to) > 0 for all i,j.By Frobenius-Perronheoryofpositivematrices see appendixn [10]),thereexistsa strictly positive eigenvaluep1(to) of M(to) whose algebraicand geometricmultiplicitiesare both one and any other eigenvaluep(to) of M(to) satisfiesIp(to)I< pi(to). This implies that the eigenvaluesXI, X2, *.., Xk of A can betaken to be such that X1 s real, pi(to) = eXitO nd(11) Xi1>ReXi for if5-l.Further f u andv satisfy(12) u*A = Xiu*, Av = Xiv,then they also satisfy(13) u*M(to) = pi(to)u*, M(to)v = pi(to)v.By the Frobenius-Perronheory of positive matricesu and v can be taken tosatisfy(14) ui > 0, v; > 0 for all i, j, ZD=iu= = 1.

2.3. Martingales.For any collectionD of randomvariableson (Q, i, P) wedenoteby a(D) the smallesta-algebracontained n 5, with respectto which allmembersof D aremeasurable.Set(15) 3t = a{X(u, w) u < t}, Y(t) = X(t)eCA, Y(t) = (Yl(t), Y(t))Then we have the followinggeneralization f the well knownmartingaleresultof the simplebranchingprocess.PROPOSITION. For every = 1, 2, , k, the amily IY(t); Tt ; t _ 01 is amartingale.PROOF.rivialusingMarkovproperty,(3) and (6). q.e.d.The same argumentalsoyields

PROPOSITION. Let t by any right eigenvectorof A with eigenvalueX. Then thefamily {I X(t)eXt; gt; t > 0} is a martingale (possibly complexvalued).We have the following mportantCOROLLARY. Letv beas in (12). Then(16) lim t-. v-X(t)e-Xt = W exists wp 1and

E(W) ? v, if X(O) ei.PROOF.Immediate since the family {v-X(t)e7't; ; ; t > 0} is a nonnegative

martingale and Fatou's lemma applies. q.e.d.



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MULTITYPE CONTINUOUS TIME MARKOV BRANCHING PROCESSES 3513. Almost sure convergence of X(t)eCX". To avoid trivial degeneracies wehenceforthexcludethe singularcase

hi(s) = Ek=1pijsj,where pij 0, Dj=i pij = 1. It is easy to see that in the singular case{X(t); t > 0} with EZo Xi(O) = 1 is essentiallya Markovchainwithstatespace{1, 2, ... , k}, transition probabilities pi jand the sojourn time in i exponentialwith parameterai.Using a recentresultof Kesten andStigum [9]on discrete ime Galton-Watsonprocesswe establish he following:

THEOREM 1. Underpositive regularity and nonsingularity(17) limto X(t, W)eXlt = W(w)u exists wp 1whereW(w) is a nonnegativenumericalvaluedrandom variableand u is defined by(12).

PROOF. Step 1. For V6 > 0 the discrete skeleton{X(nb); n = 0, 1, 2,is a discrete time Galton-Watson rocesswith mean matrixM(a). By (16) and(10), if 8no(b) > to, then M(no(5)5) = [M(5)]"O"' >> 0. Furtherthe process snonsingular.Now appealing o Kesten and Stigum'sresult [9]we conclude hatthere exists Aae 5F uch that P(A,) = 1 and X e As =* X(n5, W)eC1n1 -* W(WI, )u.It follows that lim 0 v X(nb, w)e-NI = W(w, 6) since v*u = 1.But by Corollary 1 we know limft0 v.x(t, w)eClI = W(w) exists wp 1. Thuswe have W(w, 6) = W(co),wp 1. Hence given any sequenceof biwe concludethat 3 A 313P(A) = 1and(18) weAX for ViX(n5i, w)eX 1"iW(w)u as n oo.

We writeX forXiin the remainder f this proof.STEP 2. Let D = {co:W(w) > 0}. Then on DClimv.X(t)e-t = 0 wp 1. Butvi > 0 forall i. This implieson D',

(19) Xi(t)e-t--+ 0 wp 1.It is knownthat P{D} = 0 if XI_ 0. So we shall consideronly the caseXI> 0.To establish( 17) in this casewe have to examinenowonlyD. Weshall use ( 18)to showthat on D wp 1,(20) lim inf Xj(t, c)e-t > ujW(w) for all j.However,(21) }imsup Z, viXi(t)eXt

> ( S,j lim infviXi(t)e-t) + limsupvjXj(t)e-Xt.But by (16) lim Ek=i viXi(t)e-Xt = W(co) wp 1.



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352 KRISHNA BALASUNDARAM ATHREYAThus (14), (20) and (21) =* wp 1 on D(22) limsupXj(t)e-t < ujW(w) forall j.Putting (19), (20) and (22) togetherwe get (17).It remainsonly to establish 20) andthis is ourSTEP3. By (16) P{W < oo} = 1. Let D, = {w:1/r < W(w) < r}. ThenD7 T D. To establish 20) it suffices o showV r wp 1. on D,(23) liminfXj(t)e-t > ujW(w).Fix 5 > 0 and Vt > 0 definen(t) n(t,5) by

n(t)5 _ t < (n(t) + 1)S.Fix j andlet N(t) -N(t, w) be definedas

N(t, w) = 0 if Xj(n(t)5) = 0= number among the Xj(n(t)5) that split during

[n(t)5, (n(t) + 1)51.Clearly(24) Xj(t) > Xj(n(t)5) - N(t).Weneedonly to show(25) P{w:weDr; e%fX(nS, w) > p(S) for infinitelymany n} = 0for somep( 5) ->0as 5 I 0, becausewethencoulduse (18) to get (23) from(24)and(25).To prove(25) notice

limn Xj(nl5)e -n = ujW(w) wp 1.Therefore,by Egoroff's theorem [7] Vf11 > 0, 772 > 0, 3 A e 5f such thatP(AC) < rn nd

WCEA,n> N= lXj(n5)&e -ujW(w)j p(S)} where p(5) will be specified laterand E = { we E. for infinitely many n}. Now P(E) = P(EAc) + P(EA).For n > N, P(EnA) ? PgEn,An}where(26) An = {w:w Dr, Xj(nb)e~' -ujW(W)j < 2721.But(27) P{EnAn} < Pw:w EAn,N(n5)/Xj(n5) - g(5)

> (p(b) - g(b)Xj(nt))e-n')/(Xj(n5)exna)}whereg(a) = 1 --aj

= P{a type j particle splits in [n5, (n + 1)5)1 it is alive at n5}.



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MULTITYPE CONTINUOUS TIME MARKOV BRANCHING PROCESSES 353Now on An(28) (p(8) -g(8)Xj(n5)e-A8)1Xj(n5)e-n5> (p(6) - g(8)(ujr + 72))/(ujr + X2) = g(a)if we choose p() =2g(6)(ujr + 712) where 12will be specified in (31).From (27) and (28)(29) P(EnAn) _ P{c; weAn, I(Xj(nb))Y'Zx'n" (Ur - g(W))I (?)}where

n= 1 if the rth particle among the Xj(nb) splits during [n6, (n + 1)8)0 otherwise.

But on A.(30) X.(nb)e-x"n > (ujr- - 712)Now choose X72> 0 such that(31) ujr -12-C > 0.Then by Chebychev's inequality (29), (30) and (31) yield(32) P(EnAn) N we get El' P(EnAn) < oo and hence by Borel-Cantelli we have P(EA) = 0. But P(EAC) < ql and 711s arbitrary. This estab-lishes (25). Notice p( 8) -* 0 as 8 0 since 72 does not depend on 6, r is fixed andg()O->Oas8 4, 0.q.e.d.A REMARK. Let {i- (l (i2, * * , (ik) for i = 1, 2, * , k be random variablestaking values on the k-dimensional nonnegative integer lattice with generatingfunctions respectively hi(s). Then it can be shown that(33) E(tijlogtij) < c for all i and jif and only if(34) E(Xj(t)logXj(t)IX(O) = ei) < oo forall i,j and t.Now again appealing to Kesten and Stigum's result [12]yields the following:

THEOREM 2. Either(35a) W(co) 0or(35b) P{W(w) = OIX(O) = ei} = qi, E(W(w)IX(O) = ei) v,whereqi = P{X(t) = O or some t IX(O) = ei} for i = 1, 2, *** k are theextinction probabilities and the unique root in the unit hypercube { x =(X1 , - * *Xk); 0 ?< Xi < 1} satisfying ui(x) = 0 for i = 1, 2, * - , k and



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354 KRISHNA BALASUNDARAM ATHREYAxi F 1 for at least one i. We have (35b) if and only if (33) holds. Further if (33)holds W( w) has an absolutelycontinuousdistributionon (0, oo).

4. Second moments. In the rest of this paper unless stated explicitly to thecontrary (or is clear from the context) we make the following additional assump-tions.ASSUmPTION .

(36a) xl > 0.ASSUMPTION 4.

(36b) 2hi(s)/osjask ==(9,S,...,l) < for all i, j and k.It is known that X1> 0 meais P{X(t) = 0 for some t} < 1. Harris [8] has

shown (36b) implies E(Xj(t)Xj(t) IX(O) = eC) < O for all i, j, r and t > 0.Let(37) C,Vj(t) = coV(Xi(t), Xj(t)I X(O) = er).Using (3) we can assert(38) CT(t + 8) = (I()) *Cr(t)V(8) + Z=1 mri(t)Ci(s)where CG(t) = ((Crt3(t))) and (M(s))* is the transpose of M(s). Let(39) Drtj(t) = E(Xi(t)Xj(t)IX(0) = er).From (38) it will follow that(40) Dr(t + s) = (M(s))*Dr(t)MI(s) + Z$=l mri(t)Ci(s).This yields the differential equations(41) Dr'(t) = A*Dr(t) + Dr(t)A + ZS=i mri(t)CO(0)where ' denotes differentiation.It can be verified that the unique solution of (41) is(42) Dr(t) = (M(t))*Dr(O)(11/i(t))

+ .f'M(t - ,r) *(Z$=imri(r)Ci'(0)) (M(t -r)) dr.Let t be an eigenvector of A with eigenvalue X, i.e., At = Xt. Let(43) Vr(t) - E( It X(t)12X1X(O) = er) = t*Dr(t)t.Using (42) we conclude that(44) Vr(t) = e { + Zk=i ci( ft e2a(t-)mi(r) di)where ci = t*Ci'(O) , a = Re X.Now appealing to Frobenius-Perrontheory [10]we conclude(45) limt-o M(t)e-Xlt = P = vu*.



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MULTITYPE CONTINUOUS TIME MARKOV BRANCHING PROCESSES 355Hence we get the following:

PROPOSITION 3. With the abovenotations(46a) linmtVr(t)e2at

= + It c0fo e(2a-1)rntri(r)e-\1r d- if 2a > X,7(46b) limt Vr(t)e - tt

= vr(DZ=i Ciui) if 2a =X,(46c) liMt Vr( t)e-,\ t

= Vr(DI=uici)(Xi - 2a)> if 2a < X1.This leads us to the following:

THEOREM 3. Let t be any vectorsuch that At = X) and let a = Re X be > X1/2.Then the martingale {Y(t) = t*X(t)e-t; t; t > O} satisfies (for any initial setup),supt E IY( t) 12 < 00

and hence thereexists a randomvariableY such that(47) Y(t) tends to Y wp 1 and in means square as t -- oo.

PROOF.mmediate from (43), (46a) and Doob's convergence theorem [1]. Wemake use of (42) in [2], [3].5. A Representation of X(t) - e"'tWu. In Section 3 we saw

limtX(t) e-Xt = Wu wp 1.The question arises as to what can be said about the order of magnitudeY(t) = (X(t)e-xlt - Wu). The following lemma helps in answering this.

LEMMA1. Thereexists a set A E5f such that(1) P(A) = 1.(2) wEA = for every t, thereexists randomvariables Wtej(co), j = 1, 2, ...

Xj(t, w), i- 1, 2, * , k, such that(48) X(t, w) - e"'tW(w)u = Zk=A ZX_itw) (e, - Wtt3(w)u).

PROOF. Fix a t. Then for any 1 > 0 recall the representation (3)X(t + 12Co) = Zk==i Z i(t) Xii(l, ).

NowuW(cw) = limS.+,X(s)e-Xs -limi.O, X(t + 1, co)e-1(?l).

Thusexl tuW(co) = limi, EL, EZJXi(t) Xt'(l, co)e-11.



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356 KRISHNA BALASUNDARAM ATHREYAFor a fixed t, liml .. X'j(l, co)e-^" = uWt'j(cco) exists wp 1 forj = 1, 2, ., Xj(t)i = 1, 2, *., k. Hence for each fixed t, there exists a set At e 3Ysuch thatP(At) = 1, for co8 At, Wt ii(co) exist and

e1W( = Z=1Zi(t) Wt3i(c,)and

X(t) - eXltuW() = =i Ej=i(t)ei -uwiSet A = n t rational A t. Then using the right continuity of the sample paths wehave for coe A, for every t

X(t, o) - eXltW(c() - Z fXji(tw0) ( 6i - uWit(co)). q.e.d.We now get the following theorem which says something about the order of

magnitude of Y(t) = (X(t)e-Xlt - uW).THEOREM 4. Let X1be >0 so that P{X(t) = 0 for some t} is less than one forany nontrivial initial makeup. Assume second momentsexist. Then

(49) ]iMt--x PI < Xl < W _- X2 < 00, V Y(t)e l(v.X(t))- < y}= P{O < xl -< W -< X2 < 00,}f(y/cr)

wherev is as in (12),(49a) a2 = Zk=iuiaQ,i 2 = Var (W IX(O) = ei), for i = 1, 2, k.PROOF. Let(50) Z(t) = v.Y(t)(v.X(t))-iexlt.It suffices to show(51) limt^O. E(eiOZ(t);O < X1 < W ?_ X2 < OC)

= e 2 pI < X w- W _ X2 < ?}.Now since v.X(t)e-lt -+ W a.s. it suffices to show(52) limto, E(eiOZ(t) 0 < X


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MULTITYPE CONTINUOUS TIME MARKOV BRANCHING PROCESSES 357To justify (53) we observe that from (48) we can write(54) el'tv *Y(t) (v *X(t)) = = (Xj(t)/v.X(t)) (Kj(t)) ZX=t jrwhere nlt= (vj - W't), and for r = 1, 2, * , Xj(t) are independently andidentically distributed and further independent of X(t). Now E(t1"ir) = 0,E( Iotj2) = a:2< oo.If we now appeal to the classical central limit theorem (53)yields immediately (52) since

(Xj(t)/v*X(t)) uj a.s. q.e.d.REMARKS. 1. From (48) using the argument of Theorem 4 it would seemtempting to conclude convergence of X(t) - eXtW appropriately normalized, toa multivariate normal. But such a convergence does not hold since after all

the limit random variable W is really one dimensional.2. Theorem 4 says something about the behavior in law of (X(t) - WeXlt).One would like to use (48) more strongly to assert some sample path behavior,like proving some law of the iterated logarithm etc., but this has not been doneyet. (See [11].)REFERENCES

[11 ATHREYA,RISHNA B. (1967).Limit theorems for multitype continuous time Markovbranching processes and some classical urn schemes, unpublished doctoral dis-sertation, Department of Mathematics, Stanford University.[21 ATHREYA,RISHNA . (1967).Limit theorems for multitype continuous time Markovbranching processes, I. The case of an eigenvector linear functional, (to appear).[31ATHREYA,RISHNA. (1967).Limit theorems for multitype continuous time Markovbranching processes, II. The case of an arbitrary linear functional, (to appear).[41ATHREYA,KRISHNA B. and KARLIN, SAMUEL 1966).Limit theorems for the split timesof branching processes. J. Math. Mech. 17 257-278.[51CHUNG,K. L. (1967).Markov Chains (Second Edition). Springer-Verlag, Berlin.[61 DooB, J. L. (1953).Stochastic Processes Wiley, New York.[71 HAIMos,P. R. (1959).MeasureTheory.D. Van Nostrand, Princeton.[81HARRIS,T. E. (1963).The Theory of BranchingProcesses.Springer-Verlag,Berlin.[91 IKEDA, N., NAGASAWA,M. and WATANABE,S. (1966). A construction of branchingMarkov processes. Proc. Japan Acad. 42 No. 4.

[101KARLIN, S. (1966).A First Course n Stochastic Processes. Academic Press, New York.[111KENDALL,D. G. (1966).Branching processes. J. LondonMathSoc. 1966,385-406.[121 KESTEN, H. and STIGUM,B. P. (1966). A limit theorem for multidimensional Galton-Watson process Annals of Math. Statist. 37 1211-1223.

some results on multitype continuous time markov branching processes

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