A^VÇÚO 32 ò 5 Ï 206 c 0 Chese Joural of Appled Probablty ad Statstcs Oct 206 Vol 32 No 5 pp 452-462 do: 03969/jss00-426820605002 Probablstc Meags of Numercal Characterstcs for Sgle Brth Processes LIAO Zhogwe (School of Mathematcs Su Yat-se Uversty Guagzhou 50275 Cha WANG Lgd (School of Mathematcs ad Statstcs Hea Uversty Kafeg 475004 Cha ZHANG Yuhu (School of Mathematcal Sceces Bejg Normal Uversty Bejg 00875 Cha Abstract: We cosder probablstc meags for some umercal characterstcs of sgle brth processes Some probabltes of evets such as extcto probablty returg probablty are represeted terms of these umercal characterstcs Two examples are also preseted to llustrate the value of the results Keywords: sgle brth (upwardly skp-free processes; extcto probablty; brth-death processes 200 Mathematcs Subject Classfcato: 60J60 Itroducto ad Ma Results Wag ad Yag [] preset probablstc meags for a lot of umercal characterstcs of brth-death processes such as returg probablty extcto probablty Ths paper s devoted to cosderg the correspodg problems for the sgle brth processes descrbed as follows O a probablty space (Ω F P cosder a cotuous-tme homogeeous ad rreducble Markov cha {X(t : t 0} wth trasto probablty matrx P (t = (p j (t ad state space Z + = {0 2 } We call {X(t : t 0} a sgle brth process f ts The project was supported by the Natoal Natural Scece Foudato of Cha (3003 526075 57043 Specalzed Research Fud for the Doctoral Program of Hgher Educato (20000030005 ad 985 Project from the Mstry of Educato Cha ad Scetfc Research foudato of Hea Uversty (203Y- BZR045 Correspodg author E-mal: waglgd@malbueduc Receved May 4 204 Revsed July 22 205
No 5 LIAO Z W et al: Probablstc Meags of Numercal Characterstcs for Sgle Brth Processes 453 desty matrx Q = (q j : j Z + has the followg form q 0 q 0 0 0 0 q 0 q q 2 0 0 Q = q 20 q 2 q 2 q 23 0 ( where q := q j q j q + > 0 q +j = 0 for Z + ad j 2 The matrx ( s called a sgle brth Q-matrx deduced by q j t + o(t f j < or j = + ; p j (t = q t + o(t f j = as t 0 Throughout the rest of the paper we cosder oly totally stable ad coservatve sgle brth Q-matrx: q = q = j q j < for Z + Especally f q j = 0 for 0 j 2 ad j 2 the ( s just a brth death Q-matrx Some otatos are ecessary before movg o Defe (k = k j for 0 k < (k Z + ad m 0 = q 0 m = + ( d 0 = 0 d = + ( F ( = F ( = + + k=0 + k= k=0 q (k j=0 q (k m k q (k d k F ( k 0 < The the umercal characterstcs defed below play mportat roles studyg sgle brth processes: R = m Z m = =0 =m d = sup >0 [ / d F (0 =0 =0 ] S = sup k 0 k (F (0 =0 d d To expla what the umercal characters mght mea probablty we troduce some stopg tmes respectvely e Deote the frst leapg tme ad the -th jumpg tme by η ad η η = f{t > η : X(t X(η } ; η = lm η where η 0 0 The frst httg tme ad the frst returg tme of the state are defed respectvely as follows τ = f{t > 0 : X(t = } σ = f{t η : X(t = }
454 Chese Joural of Appled Probablty ad Statstcs Vol 32 Though these umercal characterstcs may seem complex they do have explct probablstc meags ad make a postve cotrbuto towards uderstadg the process clearly Let P (A = P(A X 0 = e the codto probablty gve {X 0 = } ad E A = P (A for some measurable set A The Zhag [2] proved that m = E τ + R = E 0 η ad poted out that P 0 (σ 0 < η = Z 0 So R s the mea tme of the frst httg of the sgle brth process wth startg from 0 ad P 0 (σ 0 < η = oce /Z 0 = I [3] we see that d = E τ 0 E 0 σ 0 = /q 0 + d ad E τ 0 = (F (0 =0 d d It s easy to see that S = sup E τ 0 0 Based o the above results the followg explct crtera for several classcal problems ca be uderstood clearly (cf [2 4 7] The process s uque f ad oly f R = Assume that the Q-matrx s rreducble ad regular The the process s recurret f ad oly f Z 0 = For the regular case the process s ergodc f ad oly f d < ad the process s strogly ergodc f ad oly f S < Now we stll eed to study the probablstc meags of Z m ad defed as = wth the coveto that = =m =m > m 0 = 0 f m It wll be see later that these quattes are related to P k (τ m < τ whch s the probablty of arrvg at m alog the trajectory before reachg wth startg from k Before presetg our ma results we meto that f the sgle brth process s ergodc the the statoary dstrbuto (π ca be descrbed as (cf [8] Moreover π k = q kk+ c k c k =k [ / ] c k = sup m F (k k 0 (2 >k =k =k F (k = E k τ + E τ k 0 k < (3 It s easy to see that c k s the mea commute tme betwee k ad k + Now we preset our ma results as follows Theorem Suppose that m < The P k (τ < τ m + P k (τ m < τ = ad
No 5 LIAO Z W et al: Probablstc Meags of Numercal Characterstcs for Sgle Brth Processes 455 ( for 0 k P k (τ < τ m = Z mk P k (τ m < τ = Z mk ; ( for k > P k (τ m < τ = Z k P + (τ m < τ + Z kf (m Z mk + Moreover f the process s ergodc the P + (τ m < τ = ( c c m + + j=m+ j Z mj q( + It s easy to see that P k (τ m < τ = ad P k (τ < τ m = 0 for 0 k m P (τ m < τ = 0 ad P (τ < τ m = As for P k (σ m < τ t s obvous that P k (σ m < τ = P k (τ m < τ for k m ad P k (σ < τ m = P k (τ < τ m for k Moreover we have the followg theorem Theorem 2 Suppose that m < ( Suppose the sgle brth process s ergodc The ( P (τ m < σ = +c c m P m (σ m < τ = m+ ad P m (τ m < σ m = P m (σ m < τ m = 0 P (σ < τ m = +c c m P m (τ < σ m = m+ P k (σ m < η s the probablty of reachg m alog the trajectory through ftely may jumps wth startg from k I partcular P m (σ m < η s the probablty startg from m of returg to m alog the trajectory through ftely may jumps after leavg m whch s called a returg probablty Corollary 3 For P k (σ m < η we have f k < m; P k (σ m < η = m+ f k = m; Z m where we use the coveto that / = 0 Z mk f k > m Z m
456 Chese Joural of Appled Probablty ad Statstcs Vol 32 I practcal applcatos P k (σ 0 < η s called a extcto probablty e the probablty that there exst k dvduals tally but (through ftely may steps of trasto they fally de out (amely reach the state 0 About extcto probablty oe may also refer to [9; Chapter 9] for the case m = 0 Corollary 3 2 Proofs of the Ma Results Proof of Theorem It s easy to see that P k (τ m < τ = ad P k (τ < τ m = 0 for 0 k m To prove the remaders deote P k (τ m < τ by p k By the strog Markov property of the process for m < k we have p k = q kk+ q k p k+ + k j=0 q kj q k p j The by the coservatve property of Q-matrx ad p k = for 0 k m t follows from the above equalty that q kk+ (p k p k+ = k =m q ( k (p p + m < k (4 Deote p p + by v for 0 So we have the dfferece equato v k = k q ( q k v kk+ =m m < k < wth the boudary codtos p m = ad p = 0 By the ducto t s see that By deftos of v ad t s derved that v = v m m < (5 = p m p = =m v = v m =m = v m So v m = / ad v = / (m < Therefore t follows from p = 0 that p k = p k p = =k v = ( =k /Zm ( = =m k =m /Zm = Z mk for m < k < By the smlar argumet oe ca prove the secod part of the asserto ( Of course t s followed mmedately from the property of P k (τ m < τ + P k (τ < τ m = too
No 5 LIAO Z W et al: Probablstc Meags of Numercal Characterstcs for Sgle Brth Processes 457 To prove the asserto ( we wll dscuss frstly the relato betwee p + ad p k wth k + By (4 ad (5 t s see that The v k = ( q kk+ q ( k =m k v m v k = q kk+ k = v m + k = Defe u = v m v ( Thus oe obtas that u k = q kk+ k = q ( k v k > q ( (m k (F v m v k > q ( k u k > By the equaltes above ad the ducto t follows that u = F ( u ( Hece oe deduces that v = = ( v m u = v m F ( u = v m F ( F ( F (m v m + F ( v ( v m v Note that v = p p + = p + Furthermore t s obtaed that for k > p k = p k p = k = ( Z k = k = v = k = Z k p + + Z kf (m = ( ( (F F (m /Zm + Z k p + Z mk + v m + F ( p + By the smlar argumet or the property P k (τ m < τ + P k (τ < τ m = oe ca prove the secod part of the asserto ( Now t remas to show the asserto o the expresso of p + Usg the strog Markov property wth Theorem we have P (τ m < σ = + = + = + p + + j=0 j p j p + + q(m + j=m+ p + + q( j=m+ q ( j Z mj j Zmj Combg the above equalty wth the asserto ( Theorem 2 whch eeds oly some smple calculatos the requred asserto holds mmedately Before provg Theorem 2 we troduce the followg result (refer to [0]
458 Chese Joural of Appled Probablty ad Statstcs Vol 32 Proposto 4 dstrbuto (π The for j we have Proof of Theorem 2 Gve a ergodc Markov cha {X(t : t 0} wth the statoary P (τ j < σ = q π (E τ j + E j τ The asserto ( follows drectly from (2 (3 ad Proposto 4 by some smple calculatos By the strog Markov property ad the asserto below Theorem t turs out that P m (τ < σ m = m j=0 j P j (τ < τ m + m+ P m+ (τ < τ m = m+ P m+ (τ < τ m m+ = m+ f = m + Z mm+ = m+ Z mm+ = m+ f > m + = m+ The remaders of the asserto ( are easly obtaed Remark 5 By ducto t s ot dffcult to obta that F ( F (m m Further we get the followg equalty: Z mk Z k m < < k Hece t follows that Z Z m m < I partcular we see that Z F (0 Z 0 Z 0 for all > 0 Proof of Corollary 3 Note that τ η as almost surely wth respect to P k Hece P k (τ m < τ P k (σ m < η as for k m Combg these facts wth the assertos proved above oe gets easly the frst ad the thrd parts of the asserto By the strog Markov property ad argumet above t s see that P m (σ m < η = m j=0 = q(m m j P j (σ m < η + m+ P m+ (σ m < η = m+ Z m + q ( mm+ Z mm+ Z m I the last equalty we use the fact that Z mm+ = m =
No 5 LIAO Z W et al: Probablstc Meags of Numercal Characterstcs for Sgle Brth Processes 459 3 Examples The frst example s about the brth-death process whch s a specal class of sgle brth processes Example 6 For brth-death processes wth brth rate a ad death rate b at deoted by (a b We have these mportat quattes wth smple forms as follows m = µ[0 ]ν d = µ[ ]ν = ν ν m m 0 where µ[ k] = k µ j wth {µ } s the varat measure havg the followg form j= µ 0 = µ = b 0b b a a 2 a ( ; ad ν s aother measure related to the recurrece of the process wth ν = /µ b ( 0 I the followg we always let ν[ k] deote the term k ν j ad ν[ := ν j for some measure ( For the process we have the followg results whch ca also refer to [] Corollary 7 j= Suppose that m < For brth-death processes we have P k (τ m < τ = ad P k (τ < τ m = 0 for all 0 k m; P k (τ m < τ = 0 ad P k (τ < τ m = for all k ; ( For m < k < P k (τ < τ m = ν [m k ] ν [m ] P k(τ m < τ = ν [k ] ν [m ] ; ( P m (τ m < σ m = P m (σ m < τ m = 0 ad (v (v P m (τ < σ m = P (τ m < σ = j= (a m + b m µ m ν[m ] = P m(σ m < τ ; (a + b µ ν[m ] = P (σ < τ m ; f k < m P k (σ m < η = f k = m µ m (a m + b m ν[m ν[m k ] f k > m ν[m coveto that / = 0
460 Chese Joural of Appled Probablty ad Statstcs Vol 32 Proof By Theorems 2 ad Corollary 3 all the assertos are derved drectly except the asserto (v whch s prove as follows By the strog Markov property ad the asserto ( as well as ( we have P (τ m < σ = The proof s fshed b P + (τ m < τ + a P (τ m < τ a + b a + b = a P (τ m < τ a + b a f m = a + b = / [ a ν (a + b ] ν f m < = /[ (a + b µ Especally the extcto probablty =m =m ν ] P k (σ 0 < η = k / ν ν k =0 =0 The followg example s a exteso of the oe [8] or [] Example 8 Let + = for all 0 q 0 = b = b a 2 = a for all 2 ad q j = 0 for other j where a ad b are costats satsfyg b a > 0 k By computg we kow that {F (k } k are geeralzed Fboacc umbers for every F (k k+ = p+ + 0 k 0 p q where p = (b + b 2 + 4a/2 ad q = (b b 2 + 4a/2 Note that p > b ad < q < 0 Now ad ( p m+ p q m+ q f p ; p q p q = m q + q m+ q ( q 2 f p = ( p +2 p q+2 q f p ; p q p q m = + q + q+2 q ( q 2 f p = ( p + p q+ q f p ; p q p q d = q + q+ q ( q 2 f p =
No 5 LIAO Z W et al: Probablstc Meags of Numercal Characterstcs for Sgle Brth Processes 46 Hece t turs out that R = m =0 = e the process s always uque for all b a > 0 Moreover we get that f p ; Z m = ( p p q p + q f p < q Thus whe p (equvaletly a + b we have Z 0 = the process s recurret ad P k (σ m < η = for all k 0 Whe p < (equvaletly a + b < we have Z 0 < ad the process s traset f k < m; a + b f k = m = 0; P k (σ m < η = ( p( q f k = m > 0; + b Moreover for p we get that p( q( pk m ( pq( q k m p q f k > m P k (τ < τ m = pq(pk m q k m p k m+ + q k m+ + p q pq(p m m p m+ + m+ + p q m < k < ; (p q(p (q P 0 (τ < σ 0 = pq(p p + + + + p q 0 < ; (p q(p (q P m (τ < σ m = ( + b(pq(p m m p m+ + m+ m < + p q for p = oe obtas that k m (k m + q + qk m+ P k (τ < τ m = m ( m + q + q m+ m < k < ; ( q 2 P 0 (τ < σ 0 = m ( m + q + q m+ 0 < ; ( q 2 P m (τ < σ m = ( + b( m ( m + q + m+ m < By the way whe p = the process s ull recurret because d = Refereces [] Wag Z K Yag X Q Brth ad Death Processes ad Markov Chas [M] Bejg: Scece Press 992
462 Chese Joural of Appled Probablty ad Statstcs Vol 32 [2] Zhag J K O the geeralzed brth ad death processes (I the umeral troducto the fuctoal of tegral type ad the dstrbutos of rus ad passage tmes [J] Acta Math Sc 984 4(2: 24 259 [3] Zhag Y H Momets of the httg tme for sgle brth processes [J] J Bejg Normal Uv (Natur Sc 2003 39(4: 430 434 ( Chese [4] Che M F From Markov Chas to No-Equlbrum Partcle Systems [M] 2d ed Sgapore: World Scetfc 2004 [5] Che M F Sgle brth processes [J] Chese A Math Ser B 999 20(: 77 82 [6] Che M F Explct crtera for several types of ergodcty [J] Chese J Appl Probab Statst 200 7(2: 3 20 [7] Zhag Y H Strog ergodcty for sgle-brth processes [J] J Appl Probab 200 38(: 270 277 [8] Zhag Y H The httg tme ad statoary dstrbuto for sgle brth processes [J] J Bejg Normal Uv (Natur Sc 2004 40(2: 57 6 ( Chese [9] Aderso W J Cotuous-Tme Markov Chas: A Applcatos-Oreted Approach [M] New York: Sprger-Verlag 99 [0] Aldous D Fll J A Reversble Markov Chas ad Radom Walks o Graphs [M/OL] Preprt 204 [204-06-02] http://wwwstatberkeleyedu/ aldous/rwg/book Ralph/bookhtml [] Mao Y H Zhag Y H Expoetal ergodcty for sgle-brth processes [J] J Appl Probab 2004 4(4: 022 032 ül êavç¹â % l ( ìœæêææ 2² 50275 (àhœæêæ ÚOÆ mµ 475004 Ü{Ÿ ( ŒÆêÆ ÆÆ 00875 Á : ÄüL êa /ÏùêA x (X«ýžm! ˆž uvç ^ (JOŽü ~fƒ'êa ' c: ül ; «ývç; «L ã aò: O262