Birth and death processes with absorption

(1)

Internat. J. Math. & Math. Sc. VOL. 15 NO. 3 (1992) 469-480

469

BIRTH AND

DEATH PROCESSES WITH ABSORPTION

MOURADE. H. ISMAIL

University of South Florida

Department

of Mathematics

Tampa,

FL

33620,

U.S.A.

JEANLETESSIER

GALLIANO VALENT

LaboratoiredePhysique Thoriqueet

Hautes

Energies Unitassocie au

CNRS UA

280

UniversitParis

7, FRANCE

(Received May

8,

1991)

Abstract

Spectralmeasuresand transition probabilities of birth and deathprocesseswith _A0 _#o 0 areobtainedaslimitewhen

Ao

0+of thecorrespondingquantities. Inpa,rticula th(,caseof finitepopulationisdiscussed in full detail. Purebirth and death processesareused toderive aninequalityforDirichletpolynomials.

KEY WORDS

AND

PHRASES.

Markov

process, and inequality for

Dirichlet

polynomials.

1991

AMS

SUBJECT

CLASSIFICATION

CODES.

60J, 60J20.

1

Introduction.

A

birthand death processisastationaryMarkov process,has state space

{0,1,

andthetra]sition probabilities

p,,,,(t) (the

probabilty that the system takestime to go fronstate tostate satisfytheforwardequation

d

-p,,,,.(t)

A_p.

....

(t)

+

I+p.,,+(t)

(A

+

#,,)p

....

(t),

()

and thebackward equation

d

(t)

A.p.,+,.(t)

+

#.p._,.(t) -(A.,

+

I.)p.,.(t).

(2)

Usuallythe birthrates

{A.}

andthedeath rates

{.}

in

(1)

and

(2)

aressumedtosatisfy

t00,

A.>0d.+

>0,

n0.

(3)

Klin d

McGregor

[7]

have shown that the

p,.(/)’s

admittheintegral representation

p,,(t)

AoA

A,_,

Q(x)Q,(z)d(z),

(4)

-where

{.()}

are polynomials

orthogonal

with respect to the spectral measure

(x).

Here

(2)

470 M. ISMAIL, J. LETESSIER and G. VALENT

-.rQ,(.v) A,,Q,+,(.r)

+

y,,Q,,_,(.r) -(,,

+

y.,,)Q,,(.,.).

,,

>_

I. (5) Qo(.,’) 1. Q,(:v) (Ao+/,o

x)/.o,

ate(!_{satisfy the orthogoalitv} volatio,

(()

If probability isto I)e(’os,’vve(I,i.e. tlcl)rocsishott

[7]. [15].

1,’

Yo 0 (7)

,restI)e_{imposed. The} _interosWd_{reader may consult}_{our recent}_{survey article}

[5]

_lr_dclail. The purpose of this paper is to consider absorbingprocses with

A0

0 a,(i to

inequality for Dirichletl)olynomials. Weconsiderprocses with

A0

0 limitingca.ses

Ao

0+ of birth and death procses with

Ao

> 0. The general theory for this procedure isexpliim(I i Section2 tbr afinitestate spaceandoutlined in Section 3 foran intinitestatesl)ac(’.

I

Hc(’lio

veillustratethe procedure ofSections 2 and 3by applyingittothecazesof linear birth

processes. The dualcewherewestart with procseshaving

A0

0,Y0 nonvanishing, takig the

limit

0 0+ maybe handledin thesamewayusing the resultobtained in

[12].

Linearbirth and death processes have been studiedextensively. Anexellentretrencetoconsult is

[8].

Birth and death processes with finite state space are also called truncated procses,

[3].

Applications of truncated birth and death processes are in

[9]

and

[10]

for exemple. Karli a(i

Tava

[11]

considered birtlan(! deathprocesses withkilling. Forsuch process

p,,(t) A,l+o(t), j=i+l,

,

+o(t), j i-1,

1-(x,+,,+,)t+o(t),

0

+,

where%istherate ofkillingorabsorptionat state i. The results ofSections2 and 3

beextended tobirthand deathprocses withkillinginastraightforwardway. Theonlydifferen isthatx 0maynolongersupport adiscrete msevenifthestate

sp

isfinite.

This paperconcludwithaSection 5whereweshowhowpure birth

(or

puredeath)procses

imply thecuriousinequality

_>

)Se

-’’’ I’

(1

A.//A,)

-

>

O,

>

O, n

>

ra,

(8)

j=ra

m<l<n

ifthe

Aa’s

are distinct. The caseswhen someoftheAj’sareequal are limitingcases of thecases with distinct Aj’s.

In

Section 5 we also discuss Dirichlet series representations for p,,,,,(t) and

,,,=,,p,,.,(t)

forapure birthprocess.

As

anexampleweconsiderthecase

A,

cr(n +/3n

+

7).

Finite

state

spaces.

Recall that

(7)

is assumed throughout thissection. From (5) we see that AoQl(x)

--

-x and

(3)

BIRTH AND DEATH PROCESSES WITH ABSORPTION 471 (9)

vhere),,(x) has precisedegree

.

and

(10)

A,,o,,(.r) (A,,+/L,,-.r),,_(*)- ,,,,,,_a(.r),

,

> 1. (11) Furthermoreone canshow byinduction thatQ,,(O) for all n. n

>_

O.

In many interestingca.ses the

A,,’s

depend on a parameter aand

0

0 when o teds to a criticalvalueao, butas o o0the

X,’s

and

lt,’s,

n

>

O, havenon zerolimits. Thisisthe casefor example when

A,,

n

+

o

+

1. p. nand o -1.

It is clear that the

.’s

are birth and death process polynomials associated with birth rates

,,

anddeath rates

{t’,,

}’’.

Inthis section wetret the ceof finite state spce (finite population) becauseit is eierand

mkescertMnfeatures more transparent. Let thestate spacebe

{0,

1,...,

N}.

Define numerator

polynomiMs,

[1], [17], (,)

thesolutionof therecursion in

(5)

withthe initiM conditions

Q)(x)

o,

Q(x)

-1/o.

(12)

Itis moreconvenient inspectral analysistousethemonicpolynomials

P,.,(x)

(-1)"o...

,_lQ,(x),

P,(x)

(-1)"o...

,_Q,(x).

(3)

Itis easy toseethat

X-Ao

It 0 0

AO

X--AI--I

2 0

0

"1

X--,2--2

"’-

0

0 0 0

0

An-2 X--An--1

--#n-1

,n>O,

(14)

p,;

(.)

X--AI--I

#2 0 0 0

A1

X--A2--#2

#3 0 0

0

A2

x-Aa-/za

"’. 0 0

0 0 0

""

#n-1

0 0 0 A,_

x-A,,_-,,_

,n) 1.

(15)

Observethat the determinent representations in

(14)

and

(15)

aremeaningfulevenifsomeof

’s

and#,’svanish. Notethat

(4)

472 M. ISMAIL, J, LETESSIER and G. VALENT

Wenowassume A., O. wlicl forces tlwstatespaceI,obc

{0,

N}.

TheStieltjes transforofdp,x. th,"

nl,,’,’t.ral

,,asurcof the tiil.e proc, isivqi by

I’,+(.r)_

’[’

dt’.v(t)

(17)

l’v+

(.r)

L

r-ThezerosofPN+(x)areall real andsimpleand interlacewih lezerosof

P,+(.r). [17].

nlore

P+,(z)

lira x 1.

P+()

Thus

dttN(t

hms

Ao,

At

AN

atthezerosx0

< a

<

xNof

PN+(x)

and

N

P?v+,(x)

Ak

1,

A

>

0, 0

<

k

<

N.

(19)

Pv+,

(x) ,=0x

.,’,"

k=0

Tl,esumofthe rowsin the determinantPN+,(x)is x, hence

PN+,(0)

0. The remainingzerosof

P,x,+l(x) arepositive. Thusxo 0and (19) becomes

P+i(x)

.40

=+

A=I.

A>0, 0_<k_< N. (20)

As

A0

0.

PN+(x)

xP+(x),

hence

Ao

1. Therefore

Ak

--

0,

<_

k

<_

N and the spectral measure converges to a single point mass at x 0. Later we will see that this is

general phenomenon whenthestatespace isthe nonnegative integers and themoment problem

determined.

Recallthatfor N

>

2,

Ak

is alsogiven by

Ak

+

E

P(2",)

l’I($,_,l)

n=l ₃₌₁

Thisshows that for k

>

0,

Ao

>

0,

(21)

Therefore

[

/

a

ao,

Xo,,

+

P2(.)

n’l

(22)

Ak/Ao

I, :,)

+

_,

P(5:k)

(A,_,#,

k

>

0 ik lira x. (23)

n=2 j=2 Ao--0/

Observethat

P()

P(2-),

_<

k

<

Nand

{P(z)},

or

{ff.(x)}o

-,

isafinite family of polynomialsorthogonMwithrespecttoadiscrete (positive) probabilitymeasure

d/l(x)

suchthat

/,o__

B

+

E

[P,(x)]"

(/+,

A)

(24)

x-t

k=l 3=1

Thusweproved:

(5)

BIRTH AND DEATH PROCESSES WITH ABSORPTION 473

xdl.(.r)

Theoren and e(luatio, (-1) furnish complete information al)out the li,its of tle tra,,sitiot probabilities p (t)as

Ao

0

+.

l,,deed by writing

fff_

f(x)dlt(x)as.I’(0).4,

+

](’[.l’(.r) -.f(O)]d#(.r)

weobtain tbrm.n

>

0

p

...

(t) )’’ "")’"-’

AAoQ,,(0)Q,,(0

+

,-’t$oQ,,(.r).\oQ,,(.r)

dtt(’v) (25)

12 t. + $Ott

hellce

limp

...

(t)

A...A,,_

e-*tb,,_tCx)

(.r)dpl(x).

,.,,

>0. (26)

,’o--O+ _t _P Jo ClearlyPo,o(t) as

Ao

O.

Wealsohave, form

>

0,

fo’-"

fo

dr(x)

p,,,,o(t)

e-’Q,,,(x)dp(x)

Ao

+

/

e-’AQ"(x)

o

sothat

Similarly forn

>

O,

(27)

(28)

and

po,,,(t) O, m

>

O, as

Xo

0

+.

(29)

(30)

Therefore

lim po,(t) 5,,0.

(31)

Ao--*O+

3

Infinite

state

spaces.

Wenowextend theresultsofSection 2 to thecasewhen the statespace isthe nonnegative integers. Weassumethatthespectralmeasureisessentially unique, thatisthe momentproblemassociated withthepolynomials

{Q,,(z)}

is determined. We referthe interested readerto

[17]

for criteriafor thedeterminancyof themomentproblem.

Ifm

>

0, n

>

0 then, by letting

0

0+ in

(fi),

we see that

lim0._.0+

X-dd#(z)

d)(z)

holdsinthesensethat

f(x

dl.t(x)

f(x)d#(’)(x)

as

Ao

0

+,

(32)

foranypolynomial

f(x)

independent of

Ao.

Thisalso shows that

en}

areorthogonalwithrespect

(6)

O,,(.,.’)o,,(.r)dl;l)(x)

I21:"" l+"-h t+,, t+

>

O.

..X

.X,,

A

oredelicateargment slows tidal, tle limit relation in (32) holds if

.l"

dependso ),o but

.l’(.r)t

convergestoa polynomialas

Xo

0

+.

for all

>

0, J"

>_

0. This shows tllat

asexpected. Toseethat

dp(t)(.r)

has totalmassequalto weexpress.r intertsof go(.r), g(.+’) aud Q2(x)as

.\,Q(:,)-(o+/,,

+

,)Q,(x)+

(o

+/,o)Qo(.). (35) Thenintegrateboth sides withrespect to

dlt(x)/tt,

on

[0,

o) andapply

(6)

toget

--dlt

(x)

as

o

0

+.

,\o_l’ Thefirstmoment ofdp is

xdt,(a.)=

xQo(.r)dt,(x)

[AoQo(x)-

,XoQ,(x)]dp(x)=

o.

(37) If

f(-’)

isapolynomial ofdegreeatleast 2 then

]o

:(o,

/o

f(x)d.(x)

xf’(O)].r2dp(x)

+

f(0)

+

xdp(x)

X2

(as)

If

f(x)

isindependent of

o

the above identity

(32)

and

(37),

yields

lira

f(x)d(x)

f(O).

Xo.--*O+

(39)

Thisshowsthatas

Ao

--*0

+,

dp(x)degeneratestohavingaunit mass at z 0.

In

factweproved: Theorem 2 Letthemomentproblemcorresponding tothepolynomials

Q,,(x)

be determined. Then as

Ao

0

+,

dp(x)

degenerates toasingleton

of

unitmassatx 0 but

lim

xdp(x)

,Xo--*o+

,OPZ

where

dt)(a:)

is aprobabilit measure

suppo

on

(0, ).

Furtheothe

,

’s ave orthogonal

with resptto

dp(Z)(x).

>

0,n 0 andrn 0, n

>

O. Hereweneedtodistinguish between thecasewhen pis continuous at x 0and thecasewhenphasajumpat x 0. First assume

(o

+)

g(o-)

Ao(o) >

o,

(40)

for

₀

inaneighborhoodof

o

0,sothat

Pm,O(t)

/

(7)

BIRTH AND DZATH PROCESSES WITH ABSORPTION li _i o(1)- -f-tll

-tO

i(.r).l’-Idtllll(d’),

I1 >O,

475

(42)

il’j(;’

--.r-*dt,z)(x)

cxisls. (), _tlootl,,rlm.d if iscozztinuoust, 0tbr

Ao

0then

lieu p

....

o(1) t

-’q

(x).r-dtll(.r),

m >O. (4:1) Ao0+

[z]m cse m O, n O,since

Xo

O,rhebckward equation(])shows Lhar pu,,,( ot"I. The forvard equation (2) and simple induction provesthat,

4

Examples.

po.,,(t) bn.0, wizen

Xo

0+ (44)

Inthis section weshallconsidertowlinear birthand deathprocesses with

A,,=n+c+l, p,,,=n, (45)

A,, c(n +/3), #, n. (46)

Thereisnoloss of generalityin assuming#n n sincewerequire#o O.

Weshall alsodiscussa birthand deathprocess with rates

A.=n+o+l, t,,+ =n+c+l, n >0,

tLo=O.

(47)

Thislatercaseismorecomplicated but theanalysisisofsomeinterest.

The caseof constant birth and death rates involvesChebyshev polynomials and weleave it as anexercisefor theinterestedreader.

4.1

The process

(45).

In

thiscase

n!

L(,}(x)

,F

;x

(48)

Q,,(x)

(a

+

1)----’

a

+

where{L")(x)}

arethe

Laguerre

polynomials

[14].

Hencethe

Q,,’s

satisfytheorthogonality ,’ela-tion

Itis easy toseethat

"

)dx

Q"(z)Q(z)r(a

+

n

(a+ 1),,,

Sm,,,.

(49)

"+.,o

(-n- 1)k

)xk_

.(x)

,-.-lim

(a

x

+

1)Q"+(x)

im__

k!(a

+

j(-o

"+

1)x_

=,

k[(k-

1)

(n

+

l_xL..)_x..

Now

(14)

holds since

-we -1.

(.

+

)r(

+

)

Thus

d(1)(x)

xe-Xdx.

(50)

(8)

76 N. ISNAIL, J. LETESSIER and G. VALENT Thereforeas a --,-1 weobtain

(52)

4.2

The process

(46).

In

thecaseunder consideration,

[2, VI.3.2],

Q,(x)

m,(x/(1

c);/3,

c)/(/3),,

(3)

where the

m,’s

areMeixnerpolynomials. The orthogonality relationof theMeixner polynoials,

[2, VI.3.4],

takestheform

Q,(k(1 c))Q.i(k(1

c))c

’()

--.

(1

c)

g,.,.

(5-1)

=0

().

Now

A0

--,_{0+ in two different ways, either}_{by letting/3} ₀₊ _or_by_letting _c--,0. Weshalltreat

only the formercasebecause the lattercasebecomesacaseofa puredeathprocess.

Thusas

o

"-’0/, rd"- converge_,o to

dp()(x}

where

#()(x)

isajumpfunction,

g()(-)

0

and

()(x)

has jump

kc-l(1

-c)

at

x

(1

-c)k,

k 1,2 Clearly

#()(cx)

1, asexpected. Observe thatinthiscase _o/a blowsup atx 0as

A0

0+ whilex d"-e- converges when_X0

Ao

0+

Itis easy toseethat

(h,(x)

m,

C;2,

c

/(n

+

1)!,

(55)

henceonecanusein

(34),

(42), (44)

tocomputethetransition probabilities.

4.3

The

process

(47).

Inthiscasethe

Q,’s

arerelated toafamily ofassociated

Laguerre

polynomials

{"(x;

c)}

in

[51

by

Itwasshownin

[5]

that

(c).

(,,"(;).

(s6)

Q,.,(x)

(a

+

c

+

1).

+

1)F(1

+

c

+

a)},

(57)

where isaTricomipsifunction

[4,

Chapter

6].

Welet

A0

0+ bylettingcr -1 c. Thusin

the limitwehave

2

d/(,z’)

.l-c-z

[II(c,c.,

1;xe-i")]

-:

(58)

0#

dz

F(c

+

2)

But

the basicintegral representation

[4, 6.5.2]

shows that

q(a,

a

+

1;

z)

x-’, when

Re

a

>

0.

Therefore

z

d#(z)

xt+%

(59)

Ao#z

dx

r(c+

2)’

(9)

BIRTH AND DEATH PROCESSES WITH ABSORPTION 477 Finallywenotethat the limitingcase -cinthebirthand death process,

A,, c(,t

+

c). It,,+, n

+

c,n

>

0, ;t0 0. canbesimilaryhandled and theq,,’sareMeixner polynomials.

5

Pure

birth processes and

an

inequality.

Weapply the methode ofseparation ofvariables tofind thetransitionprobabilities. Theresult is notnewbuttheproofisofsomeinterestbecauseitalso appliestogeneralbirthand deathprocesses a.spointed out inourjointworkwith

D.

Masson

[6].

Weshallconsider first thecaseofafinitestate space

{0,

1,...,

N}

and assumethat A0,

A

AN

aredistinct.

p,.(t)

I(t)F.Q,,

(60)

antisubstitute in

(1)

and

(2)

toget

f’(t)

[A._,

F,,_,

A.F.]/F.

[AmQm+l

AmQm]

/Q

(61)

f(t)

Thus eachside intheaboveequalitymustbeaconstant. Set

f’(t)/f(t)

-x. (62)

Sincethe probabilities

p.,.(t)

alwaysliein

(0,1)

thenz

_>

0. Thisshows that

hence

Q,,+,

(1

xlAm)Q,

F._,

(A,,

x)F.IA._,

(63)

"-’

(64)

Q,,

Qo

1-I

(1

x/A.),

j=0

and similarly

AN

(65)

F.

---.

Ft

II

(1

x/Aj).

jf.+l

OnecanincorporateQ0 and

FN

intheseparation constants andfind

XN

p.,.(t)-j----t’l’l i----I

where

alp(z)

is ameasurewhich

depends

on

N. In

apurebirthprocesswemust havep.,.(t) 0

ifn

<

m,and

p,,.(t)

--.

g.,.

as -.0

+.

Thefirstrestriction implies

N

--

(67)

l’I

$--n+l i---O

a.e., hencewemaychoose

dp(x)

tobe

supported

at

{o,$,,-.-,A,}.

Thesecond restrictionis

N N m-I

(10)

wherect is the(possiblysigned)massat

At.

Thesolutionof theabove systemis

ct

(t/),,N)

II

(1

X/,X,)

-’.

O<t<N

(69)

therefore

-tAj

p

()=

II

m<l<n

(1

A,/At)-’,

n

>

m.

(70)

This provesthe inequality

(8).

Observe that therestriction N

<

oo maynow be removedfrom

(70)

provided that theseries ontherightsideconverges.

A

sufficient conditionforthis isthat

A+I

A

isbounded below bya positive number for sufficientlylargej.

Usingacompletelydifferent technique,McCarthyetal.

[13],

proved thata nontrivialDMchlet polynomial

’=1

a:

x’x’,

where

2k

E

R

and the’saredistinct, hasatmostn- zerosin

R

+.

This

shows that theseriesappering in

(8)

may have at mostn rn zeroswhen

<

0.

When

A,

a(n

+

n

+

7)

ct(n

+

a)(n

+

b),

(71)

oncancompute explicitly theproductsin

(70).

In

thiscase

II

{(At

Aj)/a}

(-1)"i-’(J

m)!(n

j)!r(n

+

j+/5

+

1)

(2j

+

)r(m

+

j

+

)

(72)

m<l<n

and

Clearly

p.,.(t)

_,

(m

+

a)._,(m

+

b),,_,(2j

+

,=, (j

m).t(n

j)!(/),,+,+,

(-1)’-’e-’b’.

(73)

(-1)S(rn

+

a)n+j(rn

+

,’,d=o

j!n!(,6),,,+:t=+.i+

(2j

+

2m

+

B)e

(-1)’(m

+

a).i(m

+

./=o

j!(fl):,,,+:t.i+,

(2j

+

2m

+

,6)e

(a+m+j,b+m+j

)

xF1

;1

+2j+2m+

A

simplification leads to

(74)

,

p,,,,,(t)

(-1)i(m

+

a).i(m

+

b).

e_.,,,/,tF

a

+

m

+

j,b

+

m

+

j ;1

j!(/5

+

2m

+

j)

/5

+

_2j

+

_2m

+

(75)

(11)

BIRTH AND DEATH PROCESSES WITH ABSORPTION 479

a.b

)

F(c)F(c-

a

b)

₍₇₍₎

toget

r(/)

0

(-1)(2j

+

2m+/)(/).+j

....

(t)

r(.,

+

.)r(,.

+

b)

(.

+

m

+

j)(b

+

m

+

j)j! (77) When a b and is an integer the abovesum can be expressed in terms of the Jacobi theta

function 0, its derivativesand its integral. Roehner simplified the expression ],,oo=,,p

....

(t)

only whenm 0, a b

1/2

(so/

1).

Snyder

[18]

consideredthecasea b (so L/= 2).

ACKNOWLEDGEMENT.

We thank

David

Masson

and

Bertrand

loehnerformanystimulating discussions during the

preparation

of this note.

Research

partially

supported

by

grant

DMS-8814026from theNationalScienceFoundation.

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