STATIONARY DISTRIBUTIONS OF THE BERNOULLI TYPE GALTON-WATSON BRANCHING PROCESS WITH IMMIGRATION

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1 Communcatons on Stochastc Analyss Vol 5, o 3 ( Serals Publcatons wwwseralsublcatonscom STATIOARY DISTRIBUTIOS OF THE BEROULLI TYPE GALTO-WATSO BRACHIG PROCESS WITH IMMIGRATIO YOSHIORI UCHIMURA AD KIMIAKI SAITÔ Abstract In ths aer, we descrbe a decomoston of the transton robablty matrx of Bernoull tye Galton-Watson branchng rocesses wth mmgraton Moreover we gve the lmtng dstrbuton of ths rocess by usng ths decomoston Introducton Branchng rocesses are mathematcal models whch are aled to the hyscal and bologcal scences The most famous branchng rocess s a Galton- Watson branchng rocess In ths aer we consder a dscrete tme Galton- Watson branchng rocess wth mmgraton It s known that ths rocess s a Markov chan whose state sace s a countably nfnte set Dscrete tme Galton- Watson branchng rocesses wth mmgraton have been descrbed n, for nstance, [, 5, 6, 7] In the recedng studes, the concrete structure of the lmtng dstrbuton and the statonary dstrbuton of the general Galton-Watson branchng rocess wth mmgraton has not been fully nvestgated The goal of ths study s to fnd these dstrbutons In ths aer, we fnd these dstrbutons for the Bernoull tye Galton-Watson branchng rocess wth mmgraton Ths s the smlest case However, t seems that even n ths smlest case, to fnd these dstrbutons s comlcated because the transton robablty matrx of ths rocess s an matrx Therefore, we ntroduce a reflectng barrer Then the transton robablty matrx s reduced to a fnte matrx For ths matrx, we construct an algorthm whch smlfes the egenolynomal (Fgure 2 These dstrbutons are derved by a matrx decomoston based on ths algorthm If π (π and ˆπ (ˆπ reresent the statonary dstrbuton and the lmtng dstrbuton, resectvely, then they can be exressed as: π ˆπ ( + λ ( (, ( Receved 2-2-9; Communcated by the edtors 2 Mathematcs Subect Classfcaton Prmary 6J8, 6F5; Secondary 6E Key words and hrases Galton-Watson branchng rocess, mmgraton, statonary dstrbuton, lmtng dstrbuton, Markov chan, generatng functon 457

2 458 YOSHIORI UCHIMURA AD KIMIAKI SAITÔ where ( s a bnomal coeffcent and {, f, λ ( k q, f k k (2 The aer s organzed as follows In secton 2, we ntroduce the defnton of the Bernoull tye Galton-Watson branchng rocess wth mmgraton In secton 3, we construct the reflectng barrer to reduce the countably nfnte state sace of ths rocess to the fnte set In secton 4, we llustrate the algorthm whch smlfes the egenolynomal of the transton robablty matrx of the Bernoull tye Galton-Watson branchng rocess In addton, the reflectng characterstc whch decdes the egenvalues of that matrx s obtaned In secton 5, we show the matrx decomoston based on the roosed algorthm llustrated n secton 4 Moreover, the lmtng dstrbuton and the statonary dstrbuton of the Bernoull tye Galton-Watson branchng rocess wth mmgraton are shown n ths secton 2 The Galton-Watson Branchng Process In ths secton, we ntroduce some defntons related to the Galton-Watoson branchng rocess Defnton 2 Let {,2, } and {} Let {X m,t ; m,t } be -valuedndeendentanddentcallydstrbuted(d randomvarables For t, Ŷt s defned by Ŷ t+ Ŷ t m X m,t, Ŷ x, (2 where x s an -valued random varable whch s ndeendent of {X m,t } The sequence {Ŷt} t s called the Galton-Watson branchng rocess Defnton 22 Let {I t ; t } be -valued d random varables For t, Y t s defned as follows: Y t+ Y t m X m,t +I t, Y x (22 The sequence {Y t } t s called the Galton-Watson branchng rocess wth mmgraton ote that the Galton-Watson branchng rocess wth mmgraton such that P(I t for each t s the same as the smle Galton-Watson branchng rocess It s known that the rocess s a Markov chan From the above defntons, we see that ts state sace s a countable nfnte set Moreover, the Galton- Watson branchng rocess has an absorbng state That s the state because clearly P(Y t+ Y t On the other hand, the general Galton-Watson branchng rocess wth mmgraton does not have any absorbng state In ths case, the state s not the absorbng state snce P(I t < We can deduce P(Y t+ Y t < from the condton P(I t <

3 STATIOARY DISTRIBUTIOS OF THE GALTO-WATSO BRACHIG PROCESS 459 Defnton 23 The Bernoull tye Galton-Watson branchng rocess wth mmgraton s defned by {Y t } t such that X m,t and I t have Bnomal dstrbutons B(, and B(,q, resectvely, e, for all m and t, P(X m,t k, f k,, f k,, otherwse, where + and q +q and P(I t k The transton robablty from to s denoted by P, e, P P(Y t+ Y t q, f k, q, f k,, otherwse, For all t, P(Y t+ Y t P(Y t Y t s deduced from Defnton 23 Hence, ths rocess s tme-homogeneous Moreover, from ths defnton, the transton robablty P can be exressed as follows: P { ( + q, f and +, q + (, otherwse, where ( a b s a bnomal coeffcent whch s defned by a! ( b!(a b!, f < b < a, a b, f a b or < a and b,, otherwse Therefore, we obtan the followng transton robablty matrx: P (23 2 n+ P P P P P 2 (24 n P n P n P n2 P nn+ Rows and columns of ths matrx are numbered from, and corresond to states The urose of ths aer s to fnd a robablty dstrbuton π (π such that πp π A robablty dstrbuton satsfyng the above equaton s called a statonary dstrbuton Moreover, f we reresent ˆπ (t P(Y t and ˆπ (t (ˆπ (t, then ˆπ ( lm t ˆπ (t f t exsts s called a lmtng dstrbuton In the case of the tme-homogeneous Markov chan, for t >, ˆπ (t can be exressed as: ˆπ (t ˆπ (t P ˆπ (t 2 P 2 ˆπ ( P t Therefore, we see that f lm t P t exsts, then so does ˆπ (

4 46 YOSHIORI UCHIMURA AD KIMIAKI SAITÔ 3 The Reflectng Barrer The Galton-Watson branchng rocess wth mmgraton has the state sace S {,,2, } Thus, P s an transton robablty matrx In ths case, to fnd the statonary dstrbuton the followng lnear smultaneous equatons wth nfntely many unknowns must be solved: P π +P π +P 2 π 2 + +P n π n + π P π +P π +P 2 π 2 + +P n π n + π P 2 π +P 22 π 2 + +P n 2 π n + π 2 P 23 π 2 + +P n 3 π n + π 3 P n n π n + π n However, ths s dffcult Therefore, we aly a reflectng barrer to the Galton- Watson branchng rocess wth mmgraton By the reflectng barrer, the state sace s reduced to a fnte set Y t Reflectng Barrer Fgure A Galton-Watson branchng rocess wth mmgraton whch has a reflectng barrer Arrows and lattce onts reresent transtons and states, resectvely In ths case, there does not exst transtons from to, for all {,,,} and > t Examle 3 For examle, the reflectng barrer can be set by the followng condtons: ( X m,t B(,

5 STATIOARY DISTRIBUTIOS OF THE GALTO-WATSO BRACHIG PROCESS 46 (2 (a If Y t <, then I t B(,q (b If Y t, then I t Then we have Y t+ { Yt m X m,t +I t, f Y t <, Yt m X m,t, f Y t Thus, fy t <, then Y t+ Y t + On the other hand, f Y t, then Y t+ Y t Moreover, the (, entry of the transton robablty matrx can be exressed as: { P, f and +, ˆP (, f and Therefore, ths Markov chan has the fnte state sace S {,,,} From Examle 3, we fnd that the state sace can be reduced to a fnte set by settng transton robabltes from sutably We call the state and the set of transton robabltes from ths state (whch s {ˆP ;,,,} the reflectng barrer and a reflectng characterstc, resectvely In ths case, the transton robablty matrx s denoted by P( Thus: P P P P P 2 P( P P P 2 P P ˆP ˆP ˆP2 ˆP ˆP The reflectng barrer whose entres of reflectng characterstc are {, f <, ˆP δ, f, s an absorbng barrer Then s clearly an absorbng state 4 Egenvalues of the Transton Probablty Matrx 4 An Algorthm for Smlfyng the Egenolynomal In ths secton, we roose an algorthm for smlfyng the egenolynomal λi P( Flowcharts of the roosed method s shown n Fgure 2 STEP, STEP2 and STEP3 n ths fgure are llustrated as follows: STEP: Let row denote the -th row of λi P( For to n, row s subtracted from row If we defne C( ( ( (, 2 ( ( (

6 462 YOSHIORI UCHIMURA AD KIMIAKI SAITÔ The Man Routne START n o n Yes STEP n n 2 STEP2 START n Yes +n%2 Yes c(,n n o n Yes STEP2 STEP3 n n ED o ED o c(,n STEP row STEP3 START row START n column n ED Yes k + Yes row column o o row row c(,n row ED Fgure 2 Flowcharts of the roosed method

7 STATIOARY DISTRIBUTIOS OF THE GALTO-WATSO BRACHIG PROCESS 463 then the matrx P( s exressed as: P( q C left (+q C rght (+R, (4 where C left ( C(, Crght ( C( and R ˆP ˆP Therefore, we aly STEP to each matrces n the equaton (4 and λi Let C STEP ( be the matrx that C left ( s rocessed by the left sde of the man routne Then the (, entry of C STEP ( can be exressed as C STEP ( ( ( 2 ( k k k+ 2 ( k k k +, f < and,, f and <,, otherwse, where s the floor functon whch s defned by x max{n Z; n x} In the above equaton, f < and, then C STEP ( ( + 2 k ( + ( k k+ 2 k2 ( k3 ( ( ( + 2 ( + ( k k+ ( k k+ 2 ( (

8 464 YOSHIORI UCHIMURA AD KIMIAKI SAITÔ Moreover, for any, the (, entry of C STEP ( can be exressed as: C STEP ( otng that we see that 2 k 2 m k k k ˆk ( +k k ( k k ( k k ( +ˆk ˆk k ( ( + + ( ( ( +2 ( +3 2 ( k2 k 2 2 ( +ˆk2 ˆk 2 ˆk 2 + m + m k2 + m ( +m+ m, ( k k ( ( k3 k2 + m ( +k k ( +k k k2 ( +k k ( ( +k k 2 2 Therefore, we obtan the followng: ( + 2 C STEP + 2, f < and + 2 (, ( 2 ( 2, f and <,, otherwse In artcular, f s an odd number, then C STEP the case of 5 we have C STEP For examle, n C STEP (5 ( ( ( ( 2 ( ( 2 ( 3 2 ( ( ( 5 5 ( 5 4 ( ( ( ( ( ( (

9 STATIOARY DISTRIBUTIOS OF THE GALTO-WATSO BRACHIG PROCESS 465 STEP2: For to n, c(n, row s subtracted from row, where c(n, s defned by {, f +n s the even number, c(n,, otherwse In the flowchart of STEP2, a%b means the remander of a/b STEP3: Let column to, column denote the -th column of λi P( For n s relaced by k column k For gven n from the man routne, let C STEP2 (n and C STEP3 (n reresent matrces whch are calculated by STEP2 and STEP3, resectvely By settng C STEP3 ( C STEP (, the (, entry of C STEP2 (n can be exressed as follows: C STEP2 (n, f < n and ; m(n (C m(n m(n STEP c(n,cstep m(n m(n, f n < ; C STEP3 (n+ ( n 2 n 2, f and < ;, otherwse;, f < n and, ( ˆm(n ˆm(n ( ˆm(n ˆm(n+ +, f n < and c(n,, ( ˆm(n ˆm(n ( ˆm(n ˆm(n, f n < and c(n,, C STEP3 (n+ ( n 2 n 2, f and <,, otherwse, where ˆm(n + 2 +m(n and m(n n Moreover, from the flowchart of STEP3, we have, f < m(n and, C STEP3 (n where Ĉ (n Snce ( f m(n, k CSTEP2 k (n, f m(n < and m(n <, Ĉ (n, f and <,, otherwse, ( k ( k k k k, ( +m(n f m(n < < n 2, k ( n 2 + n 2 ( +m(n f n 2 < ( ˆm(n ˆm(n ( ˆm(n ˆm(n ( + ˆm(n ˆm(n ( ( ˆm(n ˆm(n ( k +m(n ( m(n k k k, k ( ˆm(n ˆm(n ( k +m(n ( m(n k k k,,

10 466 YOSHIORI UCHIMURA AD KIMIAKI SAITÔ f n < and c(n,, then C STEP3 (n ( ˆm(n ˆm(n Thus C STEP3 (n s exressed as follows: C STEP3 (n ( ˆm(n ( ˆm(n ˆm(n ˆm(n, f m(n and ;, f m(n < < and c(n, ;, f m(n < < and c(n, ; Ĉ (n, f and < ;, otherwse;, f m(n and, m(n+ C (m(n+ (m(n+ STEP (n, f m(n < and m(n <, Ĉ (n, f and <,, otherwse Therefore, we obtan followng matrx: C STEP3 (n, m(n O m(n+ C STEP (n Ĉ (n Ĉ (n Ĉ m(n (n Ĉ m(n+ (n Ĉ (n where O s the zero matrx As a result of the rght sde of the man routne, we can obtan C STEP3 C STEP3 (, Ĉ Ĉ Ĉ where for {,,, }, ( Ĉ Ĉ( k ( k ( k k k Smlarly, f we aly the roosed algorthm to C rght ( and R, then ther (, entres become, resectvely,, f < m(n and,+, Ĉ, f and, Ĉ +Ĉ, f and <,, otherwse, and { k ( k ˆPk, f,, otherwse

11 STATIOARY DISTRIBUTIOS OF THE GALTO-WATSO BRACHIG PROCESS 467 Moreover, f ths method s aled to dentty matrx, t s restored Therefore, we have the followng: Theorem 4 The egenolynomal of P( can be exressed as λ q λ q λi P( λ q P P P λ+ P, (42 where and P Ĉ q Ĉ q (Ĉ +Ĉ Ĉ q Ĉ k ( k ˆPk k ( k ˆPk { ( ( k ( k k k k, f <,, f 42 Decson of Egenvalues We construct the reflectng barrer for whch P P P and P ˆλ Then the egenvalues of P( become λ,λ,,λ and λ ˆλ The reflectng characterstc {ˆP ;,,,} satsfyng ths condton can be deduced from these + equatons: Ĉ +( ( ˆP + ( ˆP + ( 2 ˆP2 + + ( ˆP + ( ˆP ˆP + ( 2 ˆP2 + + ( ˆP + ( ˆP q Ĉ +Ĉ2 +( ( 2 2 ˆP2 + + ( ˆP + ( 2ˆP q Ĉ 2 +Ĉ +( ( ˆP + ( q Ĉ +Ĉ +( ( ˆP q Ĉ + ( ˆP ˆλ Ths set of equatons can be exressed as ( Ĉ ( ( 2 ( ( Ĉ Ĉ ( ( 2 ( Ĉ 2 Ĉ ( ( ( 2 +q + 2 ( 2 2 Ĉ Ĉ 2 ( ( ( Ĉ ˆP ˆP ˆP 2 ˆP ˆP (43 ˆλ In Fgure 3, we show flowcharts of the algorthm to fnd solutons of Eq(43

12 468 YOSHIORI UCHIMURA AD KIMIAKI SAITÔ The Man Routne START n o n < Yes SUBROUTIE n n+ SUBROUTIE START n Yes o ED The +-th row of the smultaneous equatons s subtracted from ts -th row ED Fgure 3 Flowcharts of the algorthm to fnd the solutons of Eq(43 Usng ths algorthm, Eq(43 becomes Ĉ ( Ĉ ( Ĉ 2 (2 Ĉ ( +q Ĉ( Ĉ ( Ĉ( Ĉ ( Ĉ(2 2 Ĉ ( 2 2 Ĉ( Ĉ ( + ˆP ˆP ˆP 2 ˆP ˆP ( ( ˆλ ˆλ 2ˆλ, ( ( ˆλ ˆλ ( ( ( 2( where Ĉ(n s deduced from the followng recurrence relatons: Ĉ (n Ĉ Ĉ( +, f n and <,, f < n and <, Ĉ (n Ĉ(n +, f In artcular, Ĉ ( { ( k ( k ( k ( k, f <, Ĉ, f

13 STATIOARY DISTRIBUTIOS OF THE GALTO-WATSO BRACHIG PROCESS 469 Therefore, we obtan solutons of Eq(43: ˆP ( ( ˆλ q Ĉ ( ˆP ( ( ˆλ q Ĉ ( q Ĉ ( ˆP 2 ( 2( 2 ˆλ q Ĉ (2 2 q Ĉ ( ˆP ( ( ˆλ q Ĉ ( q Ĉ ( 2 2 ˆP ˆλ q Ĉ ( It means that the followng theorem holds Theorem 42 Suose the matrx P( has the reflectng barrer whose reflectng characterstc s ( ( ˆλ q Ĉ (, f, ˆP ( ( ˆλ q Ĉ ( q Ĉ (, f < <, (44 ˆλ q Ĉ (, f, where { Ĉ ( ( k ( k ( k ( k, f <, Ĉ ( (45, f Then P( has egenvalues,,, and ˆλ ote that the matrx P( wth the reflectng characterstc derved by ths method does not necessarly satsfy condtons of a robablty matrx In fact, n the case of q, q q, < q and ˆλ q, P( does not become a robablty matrx because ow, we defne ˆP ( + Q( q ( + +( q + P P P P P 2 P P P 2 P Q Q Q 2 Q (46 where {Q ;,,,} s the reflectng characterstc of Theorem 42 Moreover, the matrx Q( s defned by P P ( Q( O Q( P P P O O Q Q Q (47

14 47 YOSHIORI UCHIMURA AD KIMIAKI SAITÔ Then we have Snce P Q( P Q P Q P + P + P ++ we have lm (P Q and lm P +k, where the matrx norm s defned by lm P Q(, (48 X max a Xa and a max a k, a R k Thus the matrx Q( s a good aroxmaton for P 5 The Statonary Dstrbuton 5 The Matrx Decomoston Based on the Proosed Algorthm We consder a matrx reresentaton of the algorthm descrbed n Secton 4 It can be nferred from Theorem 4 that f ths reresentaton s aled to the matrx Q( n Secton 42, then we obtan the followng matrx: q O Λ( q (5 O We frst construct the matrx corresondng to the rocess of STEP For n gven from the man routne, ths rocess s reresented by left-multlyng by S STEP (,n I UU m(n t U m(n, where I s the dentty matrx and the matrx U s defned by U O O

15 STATIOARY DISTRIBUTIOS OF THE GALTO-WATSO BRACHIG PROCESS 47 Therefore, the (, entry of the uer matrx s exressed as S STEP (, n, f,, f m(n < and,, otherwse (52 For any X (X,, ts rows or columns are shfted by usng U In fact, UX t UX XU O X X X X X X, O X X X X O X X X X X X X X, O X X X X O X X X X X X X X O X X and X t U X X O X X X X X X X X O X X The left-multlyng Q( by S STEP (,n gves S STEP (,nq( ( I UU m(n t U m(n Q( Q( UU m(n t U m(n Q( row m(n row m(n m(n+ row m(n+ row m(n row row 2 q row row where q row ( Q Q Q

16 472 YOSHIORI UCHIMURA AD KIMIAKI SAITÔ Ths s equal to the result of STEP for gven n Therefore, the matrx reresentaton corresondng to the left sde of the man routne s gven by S STEP ( 2 k left S STEP (, 2k S STEP (, 2 2 SSTEP (, 3S STEP (, The (, entry of S STEP ( can be exressed as { S STEP ( ( +( , f + 2,, otherwse (53 ext, we consder the matrx corresondng to the rocess of STEP2 For gven n, ths rocess s reresented by the left-multlyng by the followng matrx: S STEP2 (,n I UU m(n R(, t U m(n where the matrx R(, s defned by, f and s the even number, R(, (R,, R, f and s the odd number,, otherwse Therefore, the (, entry of the matrx S STEP2 (,n s exressed as: S STEP2 (,n, f,, f m(n < and m(n s the odd number, (54, f m(n < and m(n s the even number,, otherwse Fnally, we construct the matrx corresondng to the rocess of STEP3 For gven n, t s reresented by the rght-multlyng by S STEP3 (,n t U ( m(n+ U ( m(n+ +U m(n T t U m(n, where the matrx T s defned by T O The rght-multlyng by T means the summaton from rght to left, e, for any X (X,, X X X X XT X X X X X X X X

17 STATIOARY DISTRIBUTIOS OF THE GALTO-WATSO BRACHIG PROCESS 473 Then we have S STEP3 (,n m(n O m(n + O O m(n m(n O m(n m(n O m(n O O m(n O Snce STEP2 and STEP3 are reresented by left- and rght-multly matrces, resectvely, we defne S STEP2 ( S STEP2 (, k k left S STEP2 (,m(l l S STEP2 (, S STEP2 (, 2S STEP2 (, and S STEP3 ( S STEP3 (, k k S STEP3 (,k k left S STEP3 (, S STEP3 (,S STEP3 (, ( ( ( ( (55 ( ( Then the followng holds: S STEP2 (S STEP (Q(S STEP3 ( Λ( (56

18 474 YOSHIORI UCHIMURA AD KIMIAKI SAITÔ In the equaton, S STEP2 (S STEP ( ( ( S STEP2 (,m(k k ( ( 2 ( ( ( 2( S STEP2 (,m(k k ( (l+ S STEP2 (,m(k ( l( l ( 2l( l l k ( ( ( ( 2( ( 2( 2 ( 3( 2 ( 4( 2 2 ( ( ( +( ( +2( ( 2( Thus, we see that ( S STEP2 (S STEP ( S STEP3 ( I, ( S STEP3 ( S STEP2 (S STEP ( I, namely, S STEP2 (S STEP ( s the nverse matrx of S STEP3 ( Theorem 5 The matrx Q( n (46 can be decomosed as Q( S(Λ(S ( (57 where { ( S (, f,, otherwse, { ( ( ( +, f,, otherwse, S

19 STATIOARY DISTRIBUTIOS OF THE GALTO-WATSO BRACHIG PROCESS 475 and Λ (, f, q, f +,, otherwse Moreover, we also have the followng: Theorem 52 The matrx P n Eq(24 can be decomosed as where S, Λ, and S are gven by S S P SΛS (58 { (, f,, otherwse, { ( + (, f,, otherwse,, f, Λ q, f +,, otherwse Proof We rove the entry for whch + In other cases, t s clear that P The rght-hand sde of Eq(58 s exressed as where Φ left and Φ rght are gven by SΛS Φ left +q Φ rght Φ left ( ( n O ( n n n S, ( O Φ rght ( ( n n n n S

20 476 YOSHIORI UCHIMURA AD KIMIAKI SAITÔ because SΛS ( ( O q O ( ( n ( n ( n n q S n n O ( ( q O ( ( q + ( ( n n n q (n ( n q + ( ( n n n n q + ( S ( n n n n n n q ( ( O ( ( O ( ( ( n ( n ( n +q n n ( ( n n (n S n n otng that ( +k ( ( +k! ( +k!( k!!(!! ( +k!!( +k! (! ( k!k! ( k, we see that Φ left k ( +k( k( k k ( ( ( 2+l l +l ( ( ( ( ( 2+l( ( +l +l l ( +l l ( +l ( 2+l( l l l

21 STATIOARY DISTRIBUTIOS OF THE GALTO-WATSO BRACHIG PROCESS 477 Moreover, we obtan Φ rght Therefore, ( +k( ( k+ k k k ( 2+l+( ( +l+ l+ l ( l+( l+ l +l ( ( +l ( + +l ( l+( ( +l l + l ( l ( + ( Φ left +q Φ rght +l + +l ( l+( ( + + k Thus, the roof s comleted l + ( l2+( ( +l2 l 2+ l 2 ( ( +k( k k( k +q ( ( +k( ( k+ k k ( ( ( +q + ( ( q ( +q + P +l 2 52 The Statonary Dstrbuton of the Transton Probablty Matrx P We start wth the followng Defnton 53 The generatng functon of the sequence {x t } t s defned by X(s G[x t ;s] x t s t (59 In ths defnton, we suose that s s ncluded n the regon of convergence, thus, X(s < It s known that the generatng functon has followng basc roertes Theorem 54 Consder a real sequence {x t } t Z wth x t for t < Then we have followng roertes: { ( Left Shft: G[x t+k ] s k X(s k l x ls } l (2 Rght Shft: G[x t k ] s k X(s (3 Fnal Value Theorem: f both lm s X(s and lm t x t exst, then lm s t s X(s lm x t s t k

22 478 YOSHIORI UCHIMURA AD KIMIAKI SAITÔ It s known that the generatng functon whose s to relace z s also called the Z-transform For ths theorem, we refer to [2] Let λ (k reresent the (, entry of Λ k Then we obtan λ (k+ because for any k, λ(k, f, (q λ (k + λ (k, f <,, otherwse, and λ ( {, f,, otherwse, Λ k+ Λ k Λ λ (k λ (k λ (k q λ (k λ (k λ (k q O O λ(k (q λ (k + λ (k (q λ (k + λ (k λ(k (q λ (k + λ (k O λ (k O Theorem 55 Let Λ lm k Λ k and let λ ( denote the (, entry of Λ Then we have λ (, f, k q k k, f and <,, otherwse (5 Proof We frst rove the case of By Theorem 54, for all, the generatng functon of λ (k wth resect to k can be derved as follows: G[λ (k+ s ( Λ (s λ ( ;s] G[ λ (k ;s]; Λ (s; Λ (s s Moreover, by the fnal value theorem n Theorem 54, we can obtan lm t λ(t lm s s s Λ (s lm s s s {, f,, otherwse

23 STATIOARY DISTRIBUTIOS OF THE GALTO-WATSO BRACHIG PROCESS 479 ext, weconsderthe caseof < As n the caseofthe man dagnalcomonents, for <, we can obtan G[λ (k+ s ( Λ (s λ ( ;s] G[ (q λ (k + λ (k ;s]; q Λ (s+ Λ (s; Λ (s q s sλ (s Moreover, Λ (s can be exressed as follows: Therefore, we have lm t λ(t Λ (s lm s s s { k q k k s k+ k q s k s s Λ (s lm s s k+, f,, otherwse k q s k s Fnally, t s obvous that λ ( n the case of < Therefore, Theorem 55 s satsfed From Theorem 55, we see that P SΛ S S λ ( λ ( λ ( S Therefore, the k-th entry of the lmtng dstrbuton ˆπ (ˆπ s exressed as follows: ˆπ k λ ( ( +k( k Moreover, from k P P (SΛ S (SΛS SΛ S, we see that ˆπ s also the statonary dstrbuton Therefore, we obtan the followng theorem Theorem 56 Assume that < and q Then the transton robablty matrx P gven by Equaton (24 has a unque statonary dstrbuton π (π and, for each, π s exressed as π ( + λ ( (, (5

24 48 YOSHIORI UCHIMURA AD KIMIAKI SAITÔ where λ ( {, f, k q k k Moreover, π s also the lmtng dstrbuton, f (52 References Athreya, K B and ey, P E: Branchng Processes, Dover, ew York, 24 2 Attar, R E: Lecture otes on Z-Transform, Lulu, Morrsvlle 25 3 Feller, W: An Introducton to Probablty Theory and Its Alcatons VolI, Wley, ew York, Harrs, T E: The Theory of Branchng Processes, Dover, ew York, Heathcote, C R: A Branchng Process Allowng Immgraton, J R Statst Soc B 27 ( Heathcote, C R: Correctons and Comments on th Paer A Branchng Process Allowng Immgraton, J R Statst Soc B 28 ( Seneta, E: The Statonary Dstrbuton of a Branchng Process Allowng Immgraton: A Remark on the Crtcal Case, J R Statst Soc B 3 ( Yoshnor Uchmura: Deartment of Mathematcs, Meo Unversty, agoya , JAPA E-mal address: m955@ccalumnmeo-uac Kmak Satô: Deartment of Mathematcs, Meo Unversty, agoya , JAPA E-mal address: ksato@meo-uac