ON THE NUMBER OF FAMILIES OF BRANCHING PROCESSES WITH IMMIGRATION WITH FAMILY SIZES WITHIN RANDOM INTERVAL

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1 ON THE NUMBER OF FAMILIES OF BRANCHING PROCESSES ITH IMMIGRATION ITH FAMILY SIZES ITHIN RANDOM INTERVAL Husna Hasan School of Mahemaical Sciences Universii Sains Malaysia, 8 Minden, Pulau Pinang, Malaysia husna@cs.usm.my ABSTRACT e consider he number of families in Bienayme-Galon-ason branching processes whose size is wihin a random inerval. e obain he limi heorems for he number of families in he ( n + ) s generaion whose family size wihin he random inerval for noncriical processes wih immigraion.. INTRODUCTION Consider a populaion consising of individuals able o produce offspring of he same ind. Suppose ha each individual unill by he end of is lifeime, have produce new offsprings wih probabiliy p, independenly of he number produced by any oher individual. The number of individual a ime, X, N =,,,... } is called he Bienayme-Galan-ason (BG) process. Le X, N, i = N =,,,... } be independen and idenically disribued random i variables, aing he values in he se relaion N. e define he process X, N, by he X =, X ( ) X + = Xi i = Here, X i denoes he number of descendans of he i h individual exising a ime. h Assume a he ime of birh of he generaion, ha is, a ime, here is an immigraion of Y individuals ino he populaion. Then he BG process wih immigraion (BGI) is defined by a sequence of a random variable Z which are deermined by he relaion

2 z Z ( + ) = Xi + Y + i= where he Y, Y,... are independen and idenically disribued wih common generaion funcion Y = = ( i = ) hu u PY u and hese Y ' s are independen of he random variable X } which have common i = probabiliy generaing funcion f ( u) u p funcion of Z( + ) is = X i =.Thus he probabiliy generaing Now le us consider he independen BGZ defined by where g+ u = h u g f u S Zi Z = εi i= ε i if X i ( L, U ) = if Xi ( L, U ) i =,,... for a random inerval = ( L, u). e assume X i ' cdf and independen from his random inerval (, ) and L U s are iid variables wih a common Z. If we le Z ( ) o be he number of families including he immigraing one from ime, hen he random variable S can now be inerpreed as he number of families Z (including families wih immigraing parens) living in he ( + ) s generaion who have family size wihin he random inerval ( X, X ). L U. CRITICAL PROCESSES Firs we consider he criical process. I is nown ha (see Jagers [975]) ha if f = f '' σ < h ' = µ < ', = < and hen Z / σ densiy funcion n converges in he disribuion o a random variable wih he gamma 6

3 u σ w( x) = x e u Γ σ x, u (, ) (3) Theorem : if is saisfied, hen S Z lim P z = P ( Z ( x) Z) (4) σ where Z is a random variable wih densiy funcion (3) and x = G G (5) Proof: I is clear ha for fixed u L Z and SZ is a Binomial random variable. Thus, ( = ) = Z Z = (, ) P S E P S Z Z = E ( ( x) ) ( x) e find he Laplace ransform of Z Q S : ( λqs Z ) λ Q = = Z E e = E e x ( ) λ Q = e x x P( Z = B ) = = Q ( ) λq Q E ( x) ( x) e = + P Z Q = Q B = λ ( xy ) E e dp( V g) ( = E e λ ) } z x bw by he expression lim ( ) w w ab a+ ae = e and hus bw w ab xlog ( a+ ae ) = xlog + + = w ab w+ w w w. (6) ab Thus, we have equaion (4) by aing ino accoun he limi heorem in (3). 5

4 3. NON-CRITICAL PROCESSES In he supercriical case, when he mean number of offspring < m <, hen here exiss a sequence of consiss C such ha Z( )/ C} converges wih probabiliy o a random variable V (see Jagers [975]). In his case, if ( log ), PV< = and V has an absoluely coninuous disribuion on E( log I ) =, hen PV< ( ) = Theorem : Assume < m <, hen ( Z ) E I < hen lim P S C x = P V x x where V is as menion in he limi heorem above.,. If Proof: The proof of his heorem is similar o hose for Theorem, excep we ae ino accoun he limi heorem for he supercriical case. For he subcriical case, i.e. < m < and < h ' = µ <, i is nown ha Z has a proper limi disribuion ha is lim P Z = = ρ, =,,,... (7) exis, where ρ, } saionary disribuion is ( ) and f ( u) is he generaing funcion of i is a probabiliy disribuion and he generaing funcion of his g u = h u h f u ( ) = X a ime. Theorem 3: If < m <, hen ( Z ) lim P S = = r where is a probabiliy disribuion wih generaing funcion r rµ = E g x + x u = Proof. e obain he generaing funcion of S, ( ) S Z ( µ ) = ( = E E u x x P Z Z > = = Z 6

5 = E ( x) + ( x) µ P( Z = Z > = E ( x) ( x ) µ z + }, aing ino accoun of (7). 4. ASYMPTOTICS OF MOMENTS e conclude he discussion wih he remars on he asympoic behavior of he same momen of he process S Z. Using (5), we obain he expeced value of S Z : Z Z Z E S = EE ( x) ( x) Z Z, = } = EE Z x Z, ( x)} = E Z Assuming ha Z ( ) and are independen, differeniaing, we obain E( Z ) = µ + Thus, E S = E Z E x = + E Z ( µ ) ( x) I can be shown ha for m = Var Z = Var Z ( ) + µσ ( ) + σ + τ ( ) = ( σ + τ ) + µσ. The variance of he process hen, Var S EZ ( Z [ ) ( x) ] E[ Z ( )] ( x) ( ES ) Z = + Z = Var Z ( x)} + E Z x x } which simplify o ( µ + ) E ( x) ( µ + ) E ( x) ( ) + σ + τ + µσ + ( µ + )( µ ) ( ) 5 E x. Assuming m, Z and are independen, i can be shown ha EZ m = µ + m m

6 and ( ) VarZ = m ( σ + τ ) + ( τ µ + µ ) where Thus, m m ( ) + i m S i i= = ( ) ( ) + σ µ + µ + ( EZ ) EZ S EZ m EZ m ( Z ) E S = EZ E x and m m VarS = µ + m E ( x) µ + m E Z ( x) m m m m VarZ µ m µ m E[( ( x) ) ] m m REFERENCES. Bingham N.H & Daney R.A, (974). Asympoic properies of supercriical branching processes I: Galon-ason process, Adv. Appl. Probab., 6, Harris, T.E., (963). The Theory of branching processes, Springer-Verlag. 3. Jagers P., (975). Branching Processes wih Biological Applicaion., John iley, London. 4. Paes, A.G. (97). Branching processes wih immigraion., J, Appl. Prob.8, Paes, A.G. (97). Furher resuls on he Criical Galon-ason Process wih immigraion, J. Aus. Mah. Soc. 3, Senea E., (968). The Saionary disribuion of a branching process Allowing Immigraion : A remar on he criical case, J. Royal Sais. Soc. Series B (mehodological) V3, Issue l, esolowsi, J. and Ahsanullah, M. (998). Disribuional Properies of Exceedance Saisics, Ann. Ins. Sais. Mah., 5,