Conditional Processes Induced by Birth and Death Processes

Volume 2010, Article ID 784567,24pages doi:10.1155/2010/784567

Research Article

Conditional Processes Induced by Birth and

Death Processes

Masaru Iizuka

_{and Matsuyo Tomisaki}

1_{Division of Mathematics, Kyushu Dental College, 2-6-1 Manazuru, Kokurakita-ku,}

Kitakyushu 803-8580, Japan

2_{Department of Mathematics, Faculty of Science, Nara Women’s University,}

Kita-Uoya Nishimachi, Nara 630-8506, Japan

Correspondence should be addressed to Masaru Iizuka,[email protected] Received 15 February 2010; Accepted 10 May 2010

Academic Editor: Andrew Rosalsky

Copyrightq2010 M. Iizuka and M. Tomisaki. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

For birth and death processes with finite state space, we consider stochastic processes induced by conditioning on hitting the right boundary point before hitting the left boundary point. We call the induced stochastic processes the conditional processes. We show that the conditional processes are again birth and death processes when the right boundary point is absorbing. On the other hand, it is shown that the conditional processes do not have Markov property and they are not birth and death processes when the right boundary point is reflecting.

1. Introduction

For one-dimensional diffusion processes on0,1related to diffusion models in population genetics, Ewens1considered stochastic processes induced by conditioning on hitting the boundary point 1 before hitting the other boundary point 0. The boundary points 0 and 1 are accessible and absorbing boundaries for the diffusion processes that he considered and the induced stochastic processes are again diffusion processes. Then the induced stochastic processes are referred to as the conditional diffusion processes by Ewens1 see also 2. Motivated by this work, Iizuka et al.3were concerned with one-dimensional generalized diffusion processes ODGDPs for brief on l1, l2 whose speed measures are

right-continuous and strictly increasing functions. They considered stochastic processes induced by conditioning on hitting the right boundary point l2 before hitting the left

boundary point l1. The induced stochastic processes are called the conditional processes.

boundary pointl2 is accessible with the absorbing boundary conditionAssertion 1. If the

original processxtis a one-dimensional diffusion process with the generator

L ax

2

d2

dx2 bx

d

dx, 1.1

then the conditional process x∗tinduced by conditioning on hitting l2 before hittingl1 is

again a one-dimensional diffusion process and its generator can be expressed as

L∗ ax

2

d2

dx2

bx ax s0x sx−sl1

d

dx. 1.2

Here we put

s0x exp

−2 x

c

by aydy

,

sx

_x

c

s0

ydy,

1.3

wherecis a point withl1< c < l2see4,5. On the other hand, Iizuka et al.3showed as

Theorem 2.2 that the probability distributions of the conditional processes do not satisfy the Chapman-Kolmogorov equation when the boundary pointl2is accessible with the reflecting

boundary condition. Hence the conditional processes cannot be Markov processes when the boundary pointl2is accessible with the reflecting boundary conditionAssertion 2.

An important class of ODGDPs which is used as stochastic models in various fields is that of birth and death processes. For example, Moran6introduced a birth and death process as one of fundamental stochastic models in population genetics called Moran model

we will consider this model in Section 5. However, the speed measure of any birth and death process is not a strictly increasing functionsee7and we cannot apply the results of

3to birth and death processes.

In this paper we prove that Assertions 1 and 2 hold for the case that the speed measure is a nondecreasing step function. The motivation of this paper is to investigate the properties of the conditional processes induced by conditioning on hitting the right boundary point before hitting the left boundary point when the original processes are birth and death processes. The proof of Theorem 2.2 in3is analyticalnonprobabilisticand it is not easy to see that the conditional processes do not satisfy Markov property when the right boundary point is accessible with the reflecting boundary condition. The proof presented in this paper for Assertion 2 is based on the fact that the state space is discrete. The proof is probabilistic and we can see intuitively that the conditional processes do not satisfy Markov property when the right boundary point is reflecting. It is our extra purpose to see this by considering birth and death processes.

InSection 2we state our results more precisely.Section 3is devoted to their proofs. In Section 4 we introduce a very simple birth and death process and present concrete expressions of its conditional processes considering all the boundary conditions. Finally we discuss some stochastic models in population genetics and their conditional processes in

2. Main Results

Letebe an exponentially distributed random variable with the mean 1 and let{e1,e2,e3, . . .}

be a sequence of independent copies ofe. We putτ0 0 andτk ki1ei k 1,2, . . .. For

{e1,e2,e3, . . .}, an integer N N ≥ 2,and points ai i 0,1, . . . , N such thata0 < a1 <

a2 < · · · < aN, we consider a birth and death processD Xt, Pxwith the state space

Σ {a0, a1, a2, . . . , aN}satisfying the following conditions. For ai ∈ Σand τk < t < τk1,

conditional probabilities conditional onXτk aisatisfy

PxXt ai|Xτk ai 1,

PxXτk1 ai1|Xτk ai pi,

PxXτk1 ai−1|Xτk ai qi,

PxXτk1 ai|Xτk ai 1−pi−qiri,

2.1

where 0 ≤ p0 ≤ 1, pN 0, q0 0, 0 ≤ qN ≤ 1, andpi > 0, qi > 0, ri ≥ 0 fori 1,2, . . . ,

N−1.HerePx denotes the probability measure concentrated at the event{X0 x}, that is,PxX0 x 1. The endboundarypointa0 resp.,aNis called to be absorbing or reflecting according top00resp.,qN0orp0>0resp.,qN>0.

The generatorLofDis given by

Luai pi{uai1−uai} −qi{uai−uai−1}, i1,2, . . . , N−1, 2.2

foru∈DL, whereDLis the set of all functionsuonΣsuch that

ua0 0 ifa0 is absorbing,

Lua0 p0{ua1−ua0} if a0 is reflecting,

uaN 0 ifaN is absorbing,

LuaN −qN{uaN−uaN−1} ifaN is reflecting.

2.3

Here is a proof of2.2. By means of2.1, we find that

EaiuXt EaiuXt;t < τ1 EaiuXt;τ1≤t < τ2 EaiuXt;τ2≤t,

EaiuXt;t < τ1 uaiPait < τ1 uaie−

t_,

EaiuXt;τ1 ≤t < τ2

uai1piuai−1qiuairi

1−e−te−t,

EaiuXt;τ2≤t ot ast↓0.

Therefore we obtain the following:

Luai lim t↓0

EaiuXt−uai

t

−uai uai1piuai−1qiuairi

pi{uai1−uai} −qi{uai−uai−1}

2.5

see also7.

We show that the birth and death processDcan be described as an ODGDP. We set

l1

a0 if p00,

−∞ if p0>0,

l2

aN if qN0,

∞ if qN>0,

2.6

sx

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

ρ0x, l1< x≤a0,

x−a0

a1−a0

, a0≤x≤a1,

sai

q1q2· · ·qi

p1p2· · ·pi ·

x−ai

ai1−ai

, ai≤x≤ai1, i1, . . . , N−1,

saN ρNx, aN≤x < l2,

2.7

whereρ0resp.,ρNis an increasing continuous function onl1, a0 resp.,aN, l2such that

ρ0a0 0 andρ0l1 −∞resp.,ρNaN 0 andρNl2 ∞. Further we set

mx

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

−∞, x < l1,

−1/p0, l1≤x < a0,

0, a0≤x < a1,

1/q1, a1≤x < a2,

mai−

p1p2· · ·pi−1

q1q2· · ·qi

, ai≤x < ai1, i2, . . . , N−1,

maN−

p1p2· · ·pN−1

q1q2· · ·qN

, aN≤x < l2,

∞, l2≤x.

2.8

Note that{l1 ≤ x < a0} ∅ resp.,{aN ≤ x < l2} ∅ ifp0 0resp.,qN 0. Here s is a real-valued continuous increasing function onS l1, l2, andmis a right-continuous

nondecreasing function on R. They are called the scale function and the speed measure, respectively. We setm{x} mx−mx−,mi m{ai},and si sai,i 0,1, . . . , N. We note thatm0∞resp.,mN∞ifa0resp.aNis absorbing.

For a functionf onS, we simply writefl1 resp.,fl2in place offl1 resp.,

fl2−providedfl1 resp.,fl2−exists. Further,fresp.,f−stands for the rightresp.,

leftderivative offwith respect tosif it exists, that is,fx limε↓0{fxε−fx}/{sx

We setΣ∗ Σ∩S. LetDGbe the space of all bounded continuous functionsuonS

satisfying the following conditions.

G.1There exist a functionfonΣ∗and two constantsA1, A2such that

ux A1A2{sx−sc}

c,x

sx−syfydmy, x∈S. 2.9

G.2For eachi1,2,uli 0 if|li|<∞.

Throughout this paper we denote byc an arbitrarily fixed point of Σ∗. The operator G is defined by the mapping fromu∈DGtofthat appeared in2.9. The operatorGis called the one-dimensional generalized diffusion operatorODGDO for briefwiths, m. It is known that there exists a strong Markov processD∗with the generatorG, which is called an ODGDP onSsee8,9. It is also known thatDcan be identified withD∗ see7–9. Indeed, it is easy to see thatu∈DGsatisfies the following:

uai u−ai1 uai1−uai

si1−si

, i0,1, . . . , N−1,

Guai

uai−u−ai

mi

pi{uai1−uai} −qi{uai−uai−1}, i1,2, . . . , N−1,

ua0 0 ifp00,

Gua0

ua0

m0

p0{ua1−ua0} ifp0>0,

uaN 0 ifqN0,

GuaN −

u−aN

mN −qN{uaN−uaN−1} ifqN>0.

2.10

In order to make the boundary conditions ata0andaN clear, we useDIJ andPxIJ in place ofDandPx, respectively. HereI, J ∈ {A, R},andI Aresp.,J Ameans thata0

resp.,aNis absorbingi.e.,p0 0resp.,qN 0andI Rresp.,J Rmeans thata0

resp.,aNis reflectingi.e.,p0 > 0resp.,qN > 0. It is known that there is the transition probability densitypIJ_{t, x, yofD}IJ_{with respect tom, that is,}

PxIJ

Xt ypIJt, x, ymy, t >0, x, y∈Σ∗ 2.11

see8,10.

LetΣo {a1, . . . , aN}and letσabe the first hitting time ata, that is,σa inf{t > 0 :

Xt a}. In this paper we consider stochastic processes induced by the following conditional probability:

We set

hx PxIJσaN < σa0, x∈Σ. 2.13

It is known that

hx sx−sa0 saN−sa0

sx sN

, x∈Σ 2.14

see8. We note thathis independent of boundary conditionsI, J∈ {A, R}. First we show thatQIA_{induces a birth and death process forI∈ {A, R}.}

Theorem 2.1. Assume thataNis absorbing. ThenQIAx Xt yis independent ofI∈ {A, R}, and

it is represented as

QIA_x Xt y h

y hxP

AA x

Xt y 1

sxp

AA_{t, x, ysymy, 2.15}

fort >0 andx, y∈Σo\ {aN}. FurtherQxIAinduces a birth and death processDoonΣofor which the

end pointa1is reflecting, the end pointaNis absorbing, and the generatorLois given by

Louai pi{uai1−uai} −qi{uai−uai−1}

q1· · ·qi

p1· · ·pi ·

pi

si{

uai1−uai−1}, i2, . . . , N−1,

2.16

foru∈DLo, whereDLois the set of all functionsuonΣosuch that

Loua1

p1q1

{ua2−ua1}, 2.17

uaN 0. 2.18

Theorem 2.1shows that the relation between1.1and 1.2for diffusion processes corresponds to the relation between2.2and2.16for birth and death processes. We note that the generator is given by2.17and the boundary condition2.18whenN2.

Remark 2.2. By means of2.7,

pisi1

si

pi

q1. . . qi

p1. . . pi ·

pi

si

, qisi−1 si

qi−

q1. . . qi−1

p1. . . pi−1 ·

qi

si

,

pisi1

si

qisi−1

si

piqi≤1, 1−pisi1

si −

qisi−1

si

ri.

Combining these with2.16, we find that QIA

x satisfies the following. Forx ∈ Σo\ {aN},

τk k1,2, . . .,andτk< t < τk1,

Q_xIAXt ai|Xτk ai 1,

QIA_x Xτk1 ai1|Xτk ai pi

si1

si

,

Q_xIAXτk1 ai−1 |Xτk ai qi

si−1

si

,

QIA_x Xτk1 ai|Xτk ai ri.

2.20

We turn to the case that aN is reflecting. When mx is strictly increasing, a representation ofQIR

x is given by2.11of3. We note that this representation is available even ifmxis not strictly increasing. Therefore we obtain the following representation for birth and death processes:

Q_xIRXt y s

y sxP

AA x

Xt y sN sxM

t, x, y

sN

sx

t

0

μxuNI

t−u, aN, y

du,

2.21

forI∈ {A, R},t >0,andx, y∈Σo\ {aN}. Hereμxu, Mt, x, yandNIt, x, yare given as follows. Fort >0 andx∈Σo\ {aN}, let

μxt −lim y↑aN

pAA_{t, x, a}

N−pAA

t, x, y saN−s

y . 2.22

Fort >0 andx∈Σo, let

νxt lim y↓a0

pAR_{t, x, y−p}AR_{t, x, a}

0

sy−sa0

. 2.23

It is known thatμxandνxare nonnegative density functions such as

PxIJσaN < t, σaN < σa0 _t

0

μxudu, 2.24

P_xIRσa0 < t t

0

νxudu 2.25

see8. Note thatμxis independent ofI, J∈ {A, R}andνxis independent ofI∈ {A, R}. By virtue of11, we see that

μxt

pAA_{t, x, a} N−1

sN−sN−1

, νxt

pAR_{t, x, a}

1

s1

Then we set

Mt, x, y

t

0

μxuPaARN

Xt−u ydu,

NI_{t, x, y} t

0

νxuPaIR0

Xt−u ydu.

2.27

We note thatNA_{·,·,· 0.}

The second and the third terms of the right-hand side of 2.21 come from sample path’s behavior after hitting the boundaryaN. This representation suggests thatQIRdoes not satisfy Markov property. Indeed we obtain the following theorem.

Theorem 2.3. LetI∈ {A, R}and 0< t1< t2. Then

QIR_x Xt2 y|Xt1 z

QIR_z Xt2−t1 y

2.28

does not hold for somex, y, z∈Σo. This implies thatQxIRdoes not satisfy Markov property.

This theorem is proved by applying the following simple proposition for sample path’s behavior after hitting the boundaryaN.

Proposition 2.4. LetI, J ∈ {A, R}andt >0. Then

PxIJ

Xt y, σaN < σa0

PxIJ

Xt yPxIJσaN < σa0 2.29

does not hold for somex∈Σo\ {aN}andy∈Σo.

We prove this proposition in the following section.

3. Proofs of Theorems

We use the same notations as those inSection 2.

3.1. Proof of

Theorem 2.1

Lemma 3.1. LetI, J ∈ {A, R},t >0,andx, y∈Σo\ {aN}. Then it holds true that

PxIJ

Xt y, t < σa0∧σaN

P_xAAXt y, 3.1

PxIJ

Xt yP_xAAXt yPxIJ

Xt y, t > σa0∧σaN

, 3.2

PxIJ

Xt y, t < σa0∧σaN, σaN < σa0

PxIJ

Xt y, t < σa0∧σaN

PyIJσaN < σa0

P_xAAXt yhy.

3.3

Proof ofTheorem 2.1. We assume thataNis absorbing. LetI∈ {A, R}andt >0. Then

QIA_aNXt yP_aIA_NXt y

⎧ ⎨ ⎩

1 if yaN,

0 otherwise. 3.4

Letx∈Σo\ {aN}. Then by means of2.13, and2.24,

QIA

x Xt aN

PIA

x Xt aN, σaN ≤t < σa0

PxIAσaN < σa0

1

hxP

IA

x σaN ≤t < σa0 1

hx

t

0

μxudu.

3.5

Letx, y∈Σo\ {aN}. Then by using Markov property ofDIA,2.13and3.1, we see that

QIA_x Xt y 1 hxP

IA x

Xt y, t < σaN ∧σa0, σaN < σa0

1

hxP

IA x

Xt y, t < σaN ∧σa0

P_yIAσaN < σa0

h

y hxP

AA_{Xt y.}

3.6

The formulas3.4,3.5, and3.6show thatQIA

x Xt yis independent ofI ∈ {A, R}. The formula2.15follows from2.11,2.14, and3.6.

It follows from Theorem 2.2 and Propositions 3.1 and 3.4 of12thatQIA

x induces an ODGDPDoona0, aN, the boundarya0is entrance in the sense of Fellersee8,13, the

boundaryaNis absorbing, and the generator is the ODGDOGowithso, mo, where

sox _x

c

sy−2dsy 1 sc−

1

sx, 3.7

mox

c,x

Therefore,

Gouai

uai1−uai

soai1−soai−

uai−uai−1

soai−soai−1

{moai−moai−}−1, 3.9

for a functionuonΣo\ {aN}andi2,3, . . . , N−1. By means of2.7,2.8, and3.7, we see that

{soai1−soai}{moai−moai−}

1

si − 1

si1

s2_imi si1−sisimi

si1

si

si1pi. 3.10

In the same way, we have

{soai−soai−1}{moai−moai−}

si

si−1qi

. _3.11

Therefore we get

Gouai si1

si pi{uai1−uai} −

si−1

si qi{uai−uai−1}

pi{uai1−uai} −qi{uai−uai−1}q_p1. . . qi 1. . . pi ·

pi

si{uai1−uai−1},

3.12

fori2,3, . . . , N−1. Sincea0is entrance, we see that

Goua1

ua2−ua1

soa2−soa1{

moai−moai−}−1

s2

s1

p1{ua2−ua1}

p1q1

{ua2−ua1},

3.13

by virtue of general theory on ODGDOs. Thus we find thatDois a birth and death process onΣo, the generator is given by2.16, the end pointa1is reflecting with2.17, and the end

3.2. Proof of

Theorem 2.3

We introduce the Green function corresponding toDIJ_{. ForI, J ∈ {A, R},k 1,2,andα >0,} letg_kIJ·, αbe a continuous function onSsatisfying the following properties:

g_kIJ·, α>0 onS, 3.14

gIJ₁ ·, αis nondecreasing and g₂IJ·, αis nonincreasing onS, 3.15

g₁AJa0, α 0, 3.16

g₁RJ,a0, α g₁RJa1, α−g₁RJa0, α αgRJ₁ a0, αm0, 3.17

g₂IAaN, α 0, 3.18

g₂IR,−aN, α

gIR

2 aN, α−g2IRaN−1, α

sN−sN−1 −

αg₂IRaN, αmN, 3.19

g_kIJx, α gIJ_k c, α g_kIJ,c, α{sx−sc}

α

c,x

sx−syg_kIJy, αdmy, x∈S. 3.20

Hereg_kIJ,±x, α limε↓0{gkIJx±ε, α−g IJ

k x, α}/{sx±ε−sx}. It is known that there exist such functions g_kIJ·, α, k 1,2 see 8. We set WIJ_{α g}IJ,

1 x, αg

IJ

2 x, α −

g₁IJx, αgIJ,₂ x, α. Note thatWIJ_{αis a positive number independent ofx∈S. We put}

GIJ_{α, x, yG}IJ_{α, y, xW}IJ_α−1_gIJ

1 x, αg

IJ

2

y, α, 3.21

forα > 0 andl1 < x ≤ y < l2, which is the Green function corresponding toDIJ. It is also

known that

GIJα, x, y

_∞

0

e−αtpIJt, x, ydt, 3.22

forα >0 andx, y∈Σ see8,14. First we proveProposition 2.4.

Proof ofProposition 2.4. We divide the proof into four cases.

Case 1. IJ A. SinceaNis absorbing, we find that

P_xAAXt y, σaN < σa0

P_xAAXt y, t < σa0∧σaN, σaN < σa0

P_xAAXt yhy,

3.23

by means of3.3. Since there arex, y∈Σo\ {aN}such thatx /yforN≥3,3.23shows that

LetN2. Then

P_aAA₁ Xt a2, σa2 < σa0 P AA

a1 σa2< σa0, σa2< t,

P_aAA₁ Xt a2PaAA1 σa2< σa0 P AA

a1 σa2 < σa0, σa2 < tP AA

a1 σa2 < σa0,

3.24

which imply that2.29is not valid forxa1andya2. Thus2.29does not hold for some

x, y∈Σo.

Case 2. IRandJA. By means of3.2, we also see that

P_xRAXt y, σaN < σa0

−P_xRAXt yhx

P_xAAXt yhy−hx−P_xRAXt y, t > σa0∧σaN

hx.

3.25

The right-hand side of this formula is negative ifx≥y. This implies that2.29does not hold true forx≥y.

Case 3. IAandJR. By means of3.3,

P_xARXt y, σaN < σa0

P_xAAXt yhyP_xARXt y, t > σa0∧σaN, σaN < σa0

.

3.26

Sincea0is absorbing, we get

P_xARXt y, t > σa0∧σaN, σaN < σa0

P_xARXt y, σaN < t < σa0

P_xARXt y, t > σa0∧σaN

.

3.27

Combining these equalities with3.2and3.3, we see that

P_xARXt y, σaN < σa0

−P_xARXt yhx

P_xAAXt yhy−hxP_xARXt y, σaN < t < σa0

{1−hx}.

3.28

The right-hand side of this formula is positive ifx≤y. This implies that2.29does not hold true forx≤y.

Case 4. IJ R. Suppose that2.29holds true fort >0, x∈Σo\ {aN},andy∈Σo. Then _∞

0

forα >0, x∈Σo\ {aN},andy∈Σo, where

Ht, x, y 1

my

PRR x

Xt y, σaN < σa0

−PRR x

Xt yhx. 3.30

By means of3.3,

P_xRRXt y, σaN < σa0

P_xAAXt yhyP_xRRXt y, t > σa0∧σaN, σaN < σa0

P_xAAXt yhy

t

0

P_xRRσaN ∈du, u < σa0P RR aN

Xt−u y,

3.31

forα >0 andx, y∈Σo\ {aN}.

Combining this with3.22, we see that

0 _∞

0

e−αtHt, x, ydt

GAAα, x, yhyE_xRRe−ασaN_{, σa} N < σa0

GRRα, aN, y

−GRRα, x, yhx,

3.32

forα >0 andx, y∈Σo\ {aN}, whereExIJstands for the expectation with respect toPxIJ. Here we note that3.32is valid foryaN. Indeed,

P_xRRXt aN, σaN < σa0 P RR

x Xt aN, σaN ≤t, σaN < σa0

t

0

P_xRRσaN ∈du, u < σa0P RR

aNXt−u aN.

3.33

Combining this with3.22, we see that

0 _∞

0

e−αtHt, x, aNdt

E_xRRe−ασaN_{, σa} N < σa0

GRRα, aN, aN−GRRα, x, aNhx,

3.34

Since3.32holds true forx∈Σo\ {aN}andy∈Σo, we have

0−GAAα, x, aN−1haN−1

E_xRRe−ασaN_{, σa} N < σa0

GRRα, aN, aN−GRRα, aN, aN−1

−GRRα, x, aN−GRRα, x, aN−1

hx,

3.35

forx∈Σo\ {aN}. By virtue of3.17,3.20, and3.21, we see that

GRRα, aN, aN−GRRα, aN, aN−1

WRR_α−1_gRR

1 aN, α−g1RRaN−1, α

gRR

2 aN, α

αWRRα−1sN−sN−1g2RRaN, α

a0,aN−1

g₁RRz, αdmz.

3.36

We take a pointx∈Σo\ {aN}such thatx≤aN−1. Then by virtue of3.19,3.20, and3.21,

GRR_{α, x, a}

N−GRRα, x, aN−1

WRRα−1g₁RRx, αg₂RRaN, α−g2RRaN−1, α

−αWRRα−1sN−sN−1g1RRx, α

aN−1,aN

g₂RRz, αdmz.

3.37

Thus we obtain that

0−GAAα, x, aN−1haN−1 ERRx

e−ασaN_{, σa} N < σa0

×αWRR_α−1_s

N−sN−1

N−1

l0

gRR

1 al, αgRR2 aN, αml

hxαWRRα−1sN−sN−1g1RRx, αg2RRaN, αmN.

3.38

It is known that

lim α↓0G

AA_{α, ξ, η} {sξ−sa0}

saN−s

η saN−sa0

, a0≤ξ≤η≤aN,

lim α↓0 αG

RR_{α, ξ, η} 1

ml2−ml1

, ξ, η∈S

see8,9,14,15. Therefore lettingα↓0 in3.38leads us to

0 sx

sN {−

sN−sN−1haN−1}h_mx_lsN−sN−1 2−ml1

N

l0

ml

sx

s2

N

sN−sN−12.

3.40

This contradicts the fact that the last term is positive. Thus2.29does not hold true fort >0,

x∈Σo\ {aN}, andy∈Σo.

Remark 3.2. LetN2,I J∈ {A, R}, q00, p20, andp1q11/2. Then we see that

PII

a1σa2< σa0 1 2,

P_aII₁Xt a1, σa2 < σa0 P II

a1Xt a1, σa0< σa2,

3.41

whereI∈ {A, R}. Therefore

PII

a1Xt a1, σa2 < σa0 1 2P

II

a1Xt a1 P II

a1Xt a1P II

a1σa2< σa0, 3.42

that is,2.29is valid forx y a1.Proposition 2.4implies, however, that2.29does not

hold forxa1andya2.

Proof ofTheorem 2.3. LetI ∈ {A, R}, 0< t1 < t2,x, z ∈Σo\ {aN},andy∈Σo. Then, by using Markov property ofDIR_{, we obtain that}

QIR x

Xt2 y|Xt1 z

QxIR

Xt1 z, Xt2 y

QIRx Xt1 z

PxIR

Xt1 z, Xt2 y, σaN < σa0

PxIRXt1 z, σaN < σa0

P_xAAXt1 zPzIR

Xt2−t1 y|σaN < σa0

hz

P_xIRXt1 z, σaN < t1, σaN < σa0P IR z

Xt2−t1 y

×P_xAAXt1 zhz PxIRXt1 z, σaN <t1, σaN < σa0 ₋₁

.

Therefore2.28is equivalent to the following:

P_xAAXt1 zPzIR

Xt2−t1 y|σaN < σa0

hz

P_xIRXt1 z, σaN < t1, σaN < σa0P IR z

Xt2−t1 y

PAA

x Xt1 zhz PxIRXt1 z, σaN < t1, σaN < σa0

×P_zIRXt2−t1 y|σaN < σa0

.

3.44

Again3.44is equivalent to the following:

P_xIRXt1 z, σaN < t1, σaN < σa0P IR z

Xt2−t1 y

PIR

x Xt1 z, σaN < t1, σaN < σa0P IR z

Xt2−t1 y|σaN < σa0

.

3.45

SincePIR

x Xt1 z, σaN < t1, σaN < σa0>0,3.45is equivalent to

P_zIRXt2−t1 y

P_zIRXt2−t1 y|σaN < σa0

. 3.46

However 3.46 does not hold true for some y ∈ Σo and z ∈ Σo \ {aN} by virtue of

Proposition 2.4. Thus2.28does not hold true for somex, z∈Σoandy∈Σo.

4. Examples

In this section, we consider a simple birth and death process. LetN3, aii i0,1,2,3,

q0 0, p3 0,and pi qi 1/K i 1,2, whereK ≥ 2. Forτk k 1,2, . . .given in

Section 2, the transition law of this birth and death processDsatisfies that

PxXτk1 0|Xτk 1 PxXτk1 2|Xτk 1 1

K,

PxXτk1 1|Xτk 2 PxXτk1 3|Xτk 2 1

K.

4.1

4.1. Case That the End Point 1 Is Absorbing

We first considerDAA_{, that is,}

PxXτk1 0|Xτk 0 1,

PxXτk1 3|Xτk 3 1.

Thenl10 andl23. Further2.7and2.8are reduced to

sx x, mx

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

−∞, x <0,

0, 0≤x <1,

K, 1≤x <2,

2K, 2≤x <3,

∞, 3≤x.

4.3

By virtue of11, we obtain that

P_xAAXt ypAAt, x, ymy,

pAAt, x, y 1

2Kχ{1,2}xχ{1,2}

ye−t/K −1xye−3t/K,

4.4

whereχΛξ 1resp.,χΛξ 0ifξ∈Λ resp.,ξ /∈Λ. Further by virtue ofTheorem 2.1,

QIA_x Xt y Ky x p

AA_{t, x, y, 4.5}

forx, y∈ {1,2}, whereI∈ {A, R}. Note thatQRA_x Xt yis expressed by usingpAAt, x, y. As inRemark 2.2, this induces a birth and death processDoon{1,2,3}with the transition law

QIA_x Xτk1 2|Xτk 1 1,

QIA_x Xτk1 3|Xτk 2 3 2K,

QIA

x Xτk1 1|Xτk 2 1 2K,

QIA_x Xτk1 3|Xτk 3 1.

4.6

4.2. Case That the End Point 0 Is Absorbing and the End Point 3 Is Reflecting

We next considerDAR_withq

3 1, that is,

PxXτk1 0|Xτk 0 1,

PxXτk1 2|Xτk 3 1.

For simplicity, we putK2. Thenl1 0 andl2∞. Further2.7and2.8are reduced to

sx x, mx

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

−∞, x <0,

0, 0≤x <1,

2, 1≤x <2,

4, 2≤x <3,

5, 3≤x.

4.8

By virtue of11, we obtain that

P_xARXt ypARt, x, ymy, x, y∈ {1,2,3},

pARt, x, y 1

12e

−2−√3t/2_ψAR

1 xψ1AR

y1

3e

−t_ψAR

2 xψ2AR

y

1

12e

−2√3t/2_ψAR

3 xψ3AR

y,

ψAR

1 x

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

x, x0,1,

√

3, x2,

2, x3,

ψAR

2 x

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

x, x0,1,

0, x2,

−1, x3,

ψ₃ARx

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

x, x0,1,

−√3, x2,

2, x3,

my

⎧ ⎨ ⎩

2, y1,2,

1, y3.

4.9

By means of2.21, we obtain that

QAR_x Xt y y xP

AA x

Xt y 3 xM

t, x, y

y

xp

AA_{t, x, ymy} 3

xM

t, x, ymy,

4.10

forx, y∈ {1,2}, wherepAA_{t, x, yis given by4.4withK2, and}

Mt, x, y

t

0

μxupAR

t−u,3, ydu,

μxt pAAt, x,2 1 4

e−t/2 −1xe−3t/2,

pARt,3, y 1

6

e−2−√3t/2ψ₁ARy−2e−tψ₂ARye−2√3t/2ψ₃ARy.

4.3. Case That the End Points 0 and 3 Are Reflecting

We finally considerDRR_withp

01 andq31, that is,

PxXτk1 1|Xτk 0 1,

PxXτk1 2|Xτk 3 1.

4.12

For simplicity, we putK2. Thenl1 −∞andl2∞. Further2.7and2.8are reduced to

sx x, mx

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

−1, x <0,

0, 0≤x <1,

2, 1≤x <2,

4, 2≤x <3,

5, 3≤x.

4.13

By virtue of11, we obtain that

PRR x

Xt ypRR_{t, x, ymy, x, y∈ {0,1,2,3},}

pRR_{t, x, y} 1 6ψ

RR

1 xψ1RR

y1

3e

−t/2_ψRR

2 xψ2RR

y

1

3e

−3t/2_ψRR

3 xψ3RR

y 1

6e

−2t_ψRR

4 xψ4RR

y,

ψ₁RRx 1, x0,1,2,3, ψ₂RRx

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

−1x, x0,3,

−1x−1

2 , x1,2,

ψ₃RRx

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

1, x0,3,

−1

2, x1,2,

ψ₄RRx −1x, x0,1,2,3,

my

⎧ ⎪ ⎨ ⎪ ⎩

1, y0,3,

2, y1,2.

By means of2.21, we obtain that

QRR_x Xt y y xP

AA x

Xt y 3 xM

t, x, y 3 x

_t

0

μxuNR

t−u,3, ydu

y

xp

AA_{t, x, ymy} 3

xM

t, x, ymy

3

x

_t

0

μxuNR

t−u,3, ymydu,

4.15

forx, y∈ {1,2}, wherepAA_{t, x, yis given by4.4withK 2, andMt, x, yandμ} xtare given by4.11. FurtherNRt,3, yis given as follows:

NRt, x, y

t

0

ν3upRR

t−u,0, ydu,

ν3t pARt,3,1 1

6

e−2−√3t/2−2e−te−2√3t/2,

pRRt,0, y 1

6

1−−1ye−t/2−e−3t/2 −1ye−2t.

4.16

5. Conditional Processes in Population Genetics

Here we consider two stochastic models in population genetics and their conditional processes. In this section, we use notations different from those of the previous sections to emphasize the difference between the original models and the induced models of conditional processes. We denote the conditional process byx∗t resp.,X∗twhen the original process isxt resp.,Xtas we did inSection 1.

5.1. Diffusion Model

We consider the following diffusion model for a randomly mating population consisting of

Nhaploid individuals with two typesallelesA1andA2. Letxtbe the relative frequency

ofA1at timet. Thenxtis a one-dimensional diffusion process on0,1with the generator

L x1−x

2N d2

dx2 {u21−x−u1x}

d

dx, 5.1

whereu1resp.,u2is mutation rate fromA1resp.,A2toA2resp.,A1 see4.

First we consider the case thatu1 0. The point 1 is accessible and exit boundary if

u1 0 see16. For this diffusion process, consider a stochastic process x∗tinduced by

conditioning on hitting the boundary point 1 before hitting the other boundary point 0. The induced stochastic process is again a diffusion process with the generator

L∗ x1−x

2N d2

dx2

u2

1

N

1−x d

by1.2 and Theorem 2.1 of3. Note that the effect of conditioning is that it inflates the mutation rateu2tou21/N. Ewens1considered the case thatu1u20 and the induced

diffusion process is referred to as the conditional diffusion process by Ewens1 see also

2.

Next we consider the case that 0< 2Nu1 <1. The point 1 is regular boundary in this

casesee16and we can pose various boundary conditions there. If we pose the absorbing boundary condition, then the induced process is again a diffusion process with the generator

L∗ x1−x

2N d2

dx2

u2

1

N

1−x−u1x

d

dx 5.3

by 1.2 and Theorem 2.1 of 3. On the other hand, if we pose the reflecting boundary condition as it is usually done in population genetics see 17–19, then the induced conditional process does not satisfy the Chapman-Kolmogorov equation and this process is not a diffusion process due to Theorem 2.2 of3. These results imply that we cannot use the diffusion model whose generator is given by5.3as the conditional process when we pose the reflecting boundary condition at the boundary point 1.

5.2. Moran Model

Moran 6 introduced the following birth and death process as one of the fundamental stochastic models in population genetics called continuous-time Moran modelsee4 for discrete time Moran model. We refer to this model as Moran model for brief. LetNbe the number of individuals in a haploid population with two typesA1 and A2,whereN is an

integer greater than 2. Letτ00 andτk,k1,2,· · ·be a sequence of random times introduced

inSection 2. At timeτkan individual is chosen randomly and it reproduces a new individual

k ≥ 1. The type of the newborn individual isA1resp.,A2with probability 1−ν1resp.,

1−ν2and it isA2 resp.,A1with probabilityν1 resp.,ν2if the parent isA1 resp.,A2,

where 0≤ν1, ν2≤1. Then at this timeτkan individual except newborn individual is chosen randomly to die. There is no change at timet /τk k ≥ 1. Denoting byXt the relative frequency ofA1at timet,Xtis a birth and death process on{0,1/N,2/N, . . . ,N−1/N,1}

with the transition law

P

Xτk1

j1

N |Xτk j N

1−ν1j

N−jν2

N−j2

N2 pj,

P

Xτk1

j−1

N |Xτk j N

1−ν2

N−jjν1j2

N2 qj,

P

Xτk1

j N |Xτk

j N

rj,

5.4

wherepjqjrj 10≤j ≤N. Note thatp0 ν2,q00,r0 1−ν2,pN 0,qN ν1,and

rN 1−ν1. Note also thatpj > 0 unlessν1 1 andν2 0,qj > 0 unlessν1 0 andν2 1,

andrj >0 for 0 < j < N. The processXtdoes not jump at timeτk if the types of newborn individual and the dead are the same even though a “birth and death” event occurs at time

other end point 1 is absorbingresp., reflectingwhenν10resp.,ν1 >0. Letσibe the first hitting time toii0,1.

First we consider the case thatν1ν20. This is the case without mutation and both

boundary points are absorbing with

pjqj

jN−j N2 ,

rj 1−pj−qj 1−

2jN−j

N2 .

5.5

By2.7and2.8we have

sx x,

mx

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

−∞, x <0,

0, 0≤x < 1

N, N2

N−1

N2

2N−2· · ·

N2 iN−i,

i N ≤x <

i1

N , i1, . . . , N−1,

∞, 1≤x.

5.6

ThenTheorem 2.1implies that the conditional processX∗tconditional on{σ1< σ0}is again

a birth and death process on{1/N,2/N, . . . ,N−1/N,1}with the transition law

P

X∗τk1

j1

N |X

∗_τ

k

j N

p∗_j,

P

X∗τk1

j−1

N |X

∗_τ

k

j N

q∗_j,

P

X∗τk1

j N |X

∗_τ

k

j N

r∗_j,

5.7

wherep∗_Nq∗_N0 and

p∗_j j1 j pj

j1N−j

N2 ,

q∗_j j−1 j qj

j−1N−j

N2 ,

r_j∗rj 1−

2jN−j

N2 ,

for 1≤j < N. The end point 1/Nis reflecting sincep∗₁2N−1/N2 _{>0. Note thatmxof}

the original Moran modelXtreduces to

mx

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

−∞, x <0,

0, 0≤x < 1

3, 9

2, 1 3 ≤x <

2 3, 9, 2

3 ≤x <1

∞, 1≤x,

5.9

ifN 3 and this is essentially the same as the simple birth and death process discussed in

Section 4.1.

Next we consider the case that 0< ν1<1. The boundary point 1 is reflecting. Then the

induced conditional process does not satisfy Markov property and this conditional process is not a birth and death process byTheorem 2.3.

Acknowledgments

The authors thank Thomas Nagylaki for suggesting them to consider conditional processes for Moran model in population genetics. They also thank the anonymous reviewer for comments on the previous version of this paper.

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