Asymptotic Behaviors for Critical Branching Processes with Immigration

Transcription

1 Acta Mathmatica Siica, Eglih Sri Apr., 9, Vol. 35, No. 4, pp Publihd oli: March 5, Acta Mathmatica Siica, Eglih Sri Sprigr-Vrlag GmbH Grmay & Th Editorial Offic of AMS 9 Aymptotic Bhavior for Critical Brachig Proc with Immigratio Dou Dou LI Mi ZHANG ) School of Mathmatical Scic, Laboratory of Mathmatic ad Complx Sytm, Bijig Normal Uivrity, Bijig 875, P. R. Chia lidoudou7@6.com mizhag@bu.du.c Abtract I thi papr, w ivtigat th aymptotic bhavior of th critical brachig proc with immigratio {Z, }. Firt w gt om timatio for th probability gratig fuctio of Z. Bad o it, w gt a larg dviatio for Z + /Z. Lowr ad uppr dviatio for Z ar alo tudid. A a by-product, a uppr dviatio for max i Z i i obtaid. Kyword Critical, brachig proc, immigratio, larg dviatio MR() Subjct Claificatio 6J8, 6F Itroductio Suppo {X i,,i } i a quc of o-gativ itgr-valud idpdt ad idtically ditributd (i.i.d.) radom variabl with probability gratig fuctio A(x) i a ix i. {Y, } i aothr quc of o-gativ itgr-valud i.i.d. radom variabl with probability gratig fuctio B(x) i b ix i. {X i,,i } ar idpdt with {Y, }. Dfi {Z } rcurivly a Z Z X i + Y,,Z. (.) {Z, } i calld a Galto Wato brachig proc with immigratio (GWI). Dot α : EX. Wh α>,α orα<, w hall rfr to {Z } a uprcritical, critical ad ubcritical, rpctivly. By (.), th gratig fuctio of Z ca b xprd by H (x) B[A m (x)],, (.) whr A m (x) dot th m-th itratio of th fuctio A(x) ada (x) x. Thr hav b may rarch work o th larg dviatio of Galto Wato brachig proc. Particularly, i th critical ca, wh Z ad thr i o immigratio (Y ), it i kow that { } P Z + Z >ε Z > Rcivd Sptmbr 9, 7, rvid Sptmbr 7, 8, accptd Novmbr 6, 8 Supportd by NSFC (Grat No. 873 ad 376) ) Corrpodig author. (.3)

2 538 Li D. D. ad Zhag M. Athrya ad Jagr [, p. 3, Thorm 3.] howd that if E(Z r+δ ) < for om δ>ad r, th for all ε>, thr xit q(ε) >, uch that { } P Z + Z >ε Z > q(ε) <. (.4) I [] ad [] th author timatd th uppr dviatio probabiliti of Z ad M : max k Z k udr th Cramér coditio, rpctivly. Mor xactly, i [, p. 79, Thorm] th iquality ( P (Z k) < ( + y ) + y + B wa obtaid, whr <y <R, R tad for th covrgc radiu of A() adb A ( + y ). I [, p. 6, Thorm ] th author gav that ) k ] P (M k) y [(+. y + B A for th critical Galto Wato brachig proc with immigratio, wh th fuctio A(x) adb(x) ar aalytic i th dik x < +ε for om ε>, a larg dviatio wa drivd by [8]: ( P Z bx ) y θ y dy, Γ(θ) x whr b A ( ), θ B ( ) b, x o( log )adγ( ) i th gamma fuctio. I thi papr, w hall tudy th covrgc of th typ (.4) for th critical GWI dfid by (.). Som lowr dviatio probabiliti of Z ad uppr dviatio probabiliti of Z ad M ar alo tablihd. I th proof, w hav to pay mor atttio to th chag caud by th immigratio ad d om prci timatio of th gratig fuctio of Z. W will bgi our dicuio udr th followig aumptio: (H) <a,b <, j a jj log j<, j b jj <,α, <β: B ( ) <, <γ A ( ) <. I th followig, w dfi σ β γ. W writ d O( ) if ad oly if thr xit C ad C uch that d d C C ; d if ad oly if d. C,C,... ar poitiv cotat who valu may vary from plac to plac. Th rt of th papr i orgaizd a follow. Som priary rult ar giv i Sctio. I Sctio 3 w tat th mai thorm. Sctio 4 i dvotd to th proof of th mai thorm. Priary Rult Lmma. (Athrya ad Ny [, p. 9, Thorm ]) Aum α, <γ< ad lt δ(x) γ [ A(x) x ]. Dfi h (x) δ(a m(x)) for ad h (x).th ) k x + γ A (x) h (x), x<. (.)

3 Aymptotic Bhavior for Critical GWI 539 Furthrmor, δ(x) atifi th iquality γ ( x) δ(x) ε(x), x<, a whr ε(x) :γ A(x) x ( x), which i o-icraig i x ad ε(x) a x. Lmma. (Pak [3, Thorm, ]) Udr coditio (H), w hav σ H (x) U(x), (.) whr U(x) atifi th fuctioal quatio B(x)U(A(x)) U(x). Th abov covrgc i uiform ovr compact ubt of th op uit dic. Morovr, U(x) ( x) σ, x. (.3) Dotig th powr ri rprtatio of U(x) by j μ jx j,th σ P {Z j} μ j, j. (.4) Lmma.3 (Pak [5, Thorm ]) Aum (H) hold. Lt p () j b th -tp traitio probability of {Z } from tat to j ad ν p () j j j. (i) If σ<, th ν σ U() U() d, whr U() i dfid by (.). (ii) If σ>, th ν (β γ). 3 Mai Rult Thorm 3. Aum (H) hold. For ach ε>, dfi ( A(k, ε) P Xk + Y k ), >ε (3.) whr X k k k X i. For r>σ,ifthrxitc ε > uch that A(k, ε) C ε k r for all k, th thr xit q(ε) >, uch that { } Z + σ P Z >ε Z > q(ε) <. (3.) Corollary 3. Aum (H) hold, E(X r+δ ) < ad E(Y r ) < for om δ > ad r>max{σ, }. Th(3.) hold. Thorm 3.3 Dfi J Var{ Z + Z Z > }. Aum (H) hold ad < Var(Y ) <. W hav () if σ<, th J κ σ ( + o()),

4 54 Li D. D. ad Zhag M. whr κ γ () if σ,th U() U() d +Var(Y )( μ k k k ) with {μ k } giv by (.4); (3) if σ> ad σ,th ( ) log J O ; (3.3) γ J ( + o()). (β γ) Rmark 3.4 With th rfid timat mtiod i Rmark 4., o may obtai J log γ ( + o()) ( )forσ. Thorm 3.5 Aum (H) hold. Lt k ad k o() a.th P (Z k ) C 3 (+γ k ) σ, a. Thorm 3.6 Aum (H) hold. Lt R tad for th covrgc radiu of A(x). Aum R>, k ad k o( ) a.th ( ) B (+ k k γ P (Z k ) γ ) { γ xp k γ γ + γ λk l k }( ( )) k +O (3.4) a,whrλ ρ 6γ ad ρ A ( ) <. Corollary 3.7 Aum th hypoth of Thorm 3.6 hold. Lt M max k Z k.th ( ) B (+ k k γ γ ) { γ P (M k ) xp k γ γ + γ λk l k }( ( )) k +O. (3.5) Rmark 3.8 Th right id of (3.4) ad (3.5) approximat to ( k γ )σ xp{ k γ } a. 4 Proof of Mai Rult I thi ctio w prov Thorm 3. Corollary 3.7. Firt w prt th followig propoitio. Propoitio 4. Aum (H) hold. Th for ach C 4 >, thr xit poitiv cotat C 5 ad C 6 uch that for ay < C 4, Proof ad C 5 ( + γ) σ H ( ) C6 ( + γ) σ. By Taylor formula, w kow that log x x θ (x ), x θ, x (, ), B(x) β( x) B (θ ) ( x), x θ, x (, ). Rcallig (.), w obtai [ log H (x) B(A m (x)) ] θ3 (B(A m (x)) ) [ B(A m (x))] (B(A m (x)) ) θ3

5 Aymptotic Bhavior for Critical GWI 54 whr [ ] β( A m (x)) B (θ 4 ) ( A m (x)) θ3 β ( A m (x)) + I (), I () B (θ 4 ) ( A m (x)) θ3 (B(A m (x)) ) (B(A m (x)) ), B(A m (x)) θ 3, A m (x) θ 4, x (, ). Sic A m (x) A m () mγ a m ( [, p. 74]), it i ay to how that I () i uiformly boudd for all x (, ) a. It i kow from (.), A m (x) x +γm( x) + Coqutly, log H (x) β ( A m (x)) + I () β β whr I () β m h m (x) m Hc, h m (x) m m m x +γm( x) β [ x +γm( x) x +γm( x) + I ()+I (), x +γm( x) [ { γ max A() x +γm( x) m h m (x) +γm( x) x h m (x) h m (x) +γm( x) x [ x +γm( x) m m ( A k ()), k h m (x) h m (x) +γm( x) x ]. By [3, Thorm ], m ]. h m (x) ] + I () m } m ε(a k ()) <. k i uiformly boudd for all x (, ). Furthrmor, w hav [ h m (x) +γm( x) x h m (x) ] m m γa h x m (x) m +γm( x) m( x) + γ h m(x) m h m (x) m γm m( x) + γ h m(x) m h m (x) m. m Th I () i uiformly boudd for all x (, ) a. Fially, w hav log H (x) β x +γm( x) + O()

6 54 Li D. D. ad Zhag M. uiformly for x (, ). Lt < C 4 ad x.th It ca b aily obrvd that log H ( ) β +γm Now w prov thr xit C 7, uch that +γm( + O(). (4.) ) +γm( ) +γ m ( ). I (, ) : +γ m ( ) C 7,. (4.) t To thi, ttig u(t) +γ m t( t ).Thu(t) i icraig for t>. Hc by (4.) w hav I (, ) Rcallig (4.) w obtai Sic w arriv at C 4 +C 4 γ m ( C 4 ) : C 7 <. log H ( ) β +γx dx +γm +γm + O(). +γx dx + ( + γ), σ log( + γ)+o() log H ( ) σ log( + γ)+o(). Th proof i ow complt. Rmark 4. Actually, with th hlp of a upublihd lctur ot of Prof. Vatuti, i th ca o() ( ), w may prov that H ( ) ( + γ ) σ. Proof of Thorm 3. Uig th brachig proprty, w hav { } σ Z + P Z >ε Z > A(j, ε) σ P {Z j Z > }, (4.3) j whr A(j, ε) i giv by (3.). From (.4), w hav P {Z }. Th th coditio o Z > i ot cary wh w coidr th ca. Thrfor, i th followig w oly coidr { } σ Z + P Z >ε A(j, ε) σ P {Z j}. j Nxt, w will prov a, { } σ Z + P Z >ε A(j, ε)μ j <. (4.4) j

7 Aymptotic Bhavior for Critical GWI 543 Sic A(j, ε) C ε j r,adj r (j +) r a j, th thr xit C ε uch that A(j, ε) C ε(j +) r for all j. Thrfor, l (j) : σ A(j, ε)p {Z j} σ C ε(j +) r P {Z j} : l (j). Uig (.4), w hav for j, By [7, Thorm.], l (j) C εμ j (j +) r : l(j). Th for r>σ, μ j (γ σ Γ(σ)) j σ, j. (4.5) l(j) C ε j j μ j (j +) r <. Now, uig a modificatio of th Lbgu domiatd covrgc thorm, it i ufficit to how that a, l (j) l(j), (4.6) which i quivalt to j j σ E((Z +) r ) I th followig w prov (4.7). For r>, w hav Γ(r) σ E((Z +) r ) μ j (j +) r. (4.7) j σ E( t(z +) )t r dt σ H ()( log ) r d I 3 ()+I 4 () (4.8) whr ad I 3 () I 4 () σ H ()( log ) r d, σ H ()( log ) r d. It i ay to ( log ) r d <. By Lmma., U() i boudd i [, ]. Thrfor Dfi I 3() U()( log ) r d <. (4.9) f () σ H ()( log ) r, (, ). Th f () f() :U()( log ) r.

8 544 Li D. D. ad Zhag M. From Propoitio 4., w kow that for t [, ), thr xit N ad C 8 uch that for >N, Hc, for all >N. It i ot difficult to ad for r>σ, H (t) C 8 ( γlog t) σ. f () C 8 σ ( γlog ) σ ( log ) r : g () g () C 8 ( γ log ) σ ( log ) r : g(), g()d C 8 (γt) σ t r t dt <. Uig th modificatio of domiatd covrgc thorm, w hav f ()d f()d,. (4.) By a chag of variabl u log, th right id of (4.) tur out to b U( u )u r u du, which i fiit by uig (.3). Hc, w obtai I 4() Combiig (4.8), (4.9) with (4.), w hav which yild Clarly, Γ(r) σ E((Z +) r ) j l (j) C ε Γ(r) C ε Γ(r) f()d <. (4.) f()d <, f()d f()d <. l(j). j, Thu w gt (4.6), ad th (4.4) hold. Th proof i compltd. Proof of Corollary 3. By Markov iquality, w hav ( A(k, ε) P Xk + Y ) k >ε E( k( X k + Y k )) r ε r k r. Uig th aumptio ad [6, p., Sctio 9.9], w obtai ( ( k C ε upe X k + Y )) r k k <. Th thr xit a cotat C ε uch that A(k, ε) C ε k r for all k.

9 Aymptotic Bhavior for Critical GWI 545 Proof of Thorm 3.3 Var { Z+ Z Firt w dicu k hall prov W kow that k Lt p k P (Z k Z > ). By dirct calculatio, w hav } Z > Var(X ) p k k +Var(Y ) p k k. (4.) k k p k k. Th ca σ hav b giv by Lmma.3. For σ,w k p k k ( ) log O. (4.3) p k k H (x) H () x( H ()) dx : I 5()+I 6 (), whr H (x) H () I 5 () x( H ()) dx, ad H (x) H () I 6 () x( H ()) dx, with log. Uig Lmma. w hav that H () x( H ()) dx H () log ( H ()) μ σ log,. Lt I 6 () H (x) x dx. Nxt w coidr th ordr of I 6 (). By Propoitio 4., I 6 () ( ) log H ( θ log )dθ O. Morovr, oticig that H (x) H () E(x Z Z > ) x( H ()) i o-dcraig i x, th by th dfiitio of I 5 () ad Propoitio 4., w obtai I 5 () H ( ( ) ) log H () O. Thu (4.3) hold. Now w tur to timat (i) If σ<, by (4.4), (.4) ad (4.5), σ ν Th w hav ν : k k ν σ k p k k. σ p k k k μ k k <. μ k,. (4.4) k

10 546 Li D. D. ad Zhag M. (ii) If σ>, firt it i kow that Γ() ν E( tz Z > )tdt ( E( tz,z > )tdt + P (Z > ) : P (Z > ) (I 7()+I 8 ()). By a chag of variabl t,whav I 7 () : (H ( ) H ())d I ( ) (H ( ) H ())d q ()d. Uig Propoitio 4., thr xit C 9 uch that Clarly, for σ>, q () C 9 ( + γ) σ : v(). v()d <. Thrfor, uig th domiatd covrgc thorm, w hav whr by [4, Thorm 3], For I 8 (), I 8 () I 7() q()d <, q() : q () (+γ) σ. E( tz I (Z ))tdt + : K ()+K (). By (.4) ad th Lbgu domiatd covrgc thorm, K () P (Z ) t tdt, ) E( tz,z > )tdt E( tz I (Z ))tdt. Mawhil, by (.) ad th Lbgu domiatd covrgc thorm, Th, w gt K () E( t(z ) I (Z )) t tdt E( t Z ) t tdt H ( t ) t tdt, I 8()..

11 Aymptotic Bhavior for Critical GWI 547 Lt W gt C q()d. ν C,. (4.5) Collctig (4.) (4.5) ad combiig with Lmma.3, w obtai th rult. Proof of Thorm 3.5 For all >, w hav P (Z k )P( Z k ) E( Z ) k H ( ) k. Lttig k ad applyig Propoitio 4., w hav P (Z k ) C 3 (+γ k ) σ. Proof of Thorm 3.6 Lt <y <R. Lt th quc y b dfid by th quatio A( + y + )+y. It i ot difficult to that A( + y) +y for y. Thrfor, th quc y dcra. Th log H ( + y ) log B( + y m ) log B( + y i ) [ B( + y i ) θ 5 [B (θ 6 )y i B (θ 6 ) B ( + y ) θ 5 y i + C y i (B( + y i ) ) ] whr θ 5 B( + y i ), θ 6 +y i,i, ad C ( B ( ) B(+y ) ). Thrfor, { } H ( + y ) xp B ( + y ) y i + C. By [, Lmma ], Sttig y i γ l( + γy )+O(y ), i y k γ γ, ] y i, y i yi O(y ). i

12 548 Li D. D. ad Zhag M. w obtai Not that for all y>, H ( + y ) H ( + y) ( ) B (+ k k γ γ ) γ. (4.6) γ P (Z j)( + y) j j ( + y) k P {Z k }. (4.7) By [] { ( + y ) k xp k γ + γ λk l k }( +O ( k )). (4.8) Th thorm i provd by combiig (4.6), (4.7) with (4.8). Proof of Corollary 3.7 For vry t> w dfi D (t) tz,. It i ay to chck that {D } i a ubmartigal with rpct to th atural σ-algbra gratd by {Z }.Byth Doob iquality, ( P (M k) P max D i(t) tk) H ( t ) i tk. Dfi y a i th proof of Thorm 3.6 ad lt t log(+y ). Th by th proof of Thorm 3.6, w ca obtai th dird rult. Ackowldgmt W ar gratful to Profor Vatuti for hlpful uggtio o Propoitio 4.. Thak ar alo giv to th rfr for thir commt ad uggtio. Rfrc [] Athrya, K. B., Ny, P. E.: Brachig Proc, Sprigr-Vrlag, Brli, 97 [] Athrya, K. B., Jagr, P.: Claical ad Modr Brachig Proc, Sprigr, Brli, 999 [3] Athrya, K. B.: Larg dviatio rat for brachig proc I. igl typ ca. A. Appl. Probab., 4(3), (994) [4] Flichma, K., Wachtl, V.: Lowr dviatio probabiliti for uprcritical Galto Wato proc. A. I. H. Poicar, 39(4), (7) [5] Liu, J. N., Zhag, M.: Larg dviatio for uprcritical brachig proc with immigratio. Acta Math. Siica, 3(8), (6) [6] Li, Z. Y., Bai, Z. D.: Probability Iqualiti, Scic Pr, Bijig ad Sprigr-Vrlag, Brli, [7] Mlli, B.: Local it thorm for th critical Galto Wato proc with immigratio. Rv. Colomb. Mat., 6, 3 56 (98b) [8] Makarov, G. D.: Larg Dviatio for brachig proc with immigratio. Math. Not., 3, 4 4 (98) [9] Nagav, A. V.: O timatig th xpctd umbr of dirct dcdat of a particl i a brachig proc. Thory Probab. Appl.,, 34 3 (967) [] Nagav, S. V., Vakhtl, V. I.: Limit thorm for probabiliti of larg dviatio of a Galto Wato proc. Dicrt Math. Appl., 3(), 6 (3) [] Nagav, S. V., Vakhruhv, N. V.: Etimatio of probabiliti of larg dviatio for a critical Galto Wato proc. Thory Probab. Appl.,, 79 8 (976)

13 Aymptotic Bhavior for Critical GWI 549 [] Nagav, S. V., Vakhtl, V. I.: Probability iqualiti for a critical Galto Wato proc. Thory Probab. Appl., 5, 5 47 (6) [3] Pak, A. G.: Furthr rult o th critical Galto Wato proc with immigratio. J. Aut. Math. Soc., 3, 77 9 (97) [4] Pak, A. G.: O th critical Galto Wato proc with immigratio. J. Aut. Math. Soc.,, (969) [5] Pak, A. G.: No-paramtric timatio i th Galto Wato proc. Math. Bio., 6(), 8 (975)