On the limit structure of continuous-time Markov branching process Текст научной статьи по специальности «Математика»

УДК 519.218.2

On the Limit Structure of Continuous-time Markov Branching Process

Azam А. Imomov*

State Testing Center CMRU Institute of Mathematics, National University of Uzbekistan Durmon yuli, 29, Tashkent, 100125

Uzbekistan

Received 23.08.2016, received in revised form 10.09.2016, accepted 01.11.2016 We study the limiting probability function of continuous-time Markov Branching Processes conditioned to be never extinct. Hereupon we obtain a new stochastic population process called the Markov Q-Process. The principal aim is to investigate structural and asymptotic properties of the Markov Q-Process, also we study transition functions of this process and their convergence to stationary measures.

Keywords: Markov Branching Process, Markov Q-process, transition function; invariant measures. DOI: 10.17516/1997-1397-2017-10-1-117-127.

1. Introduction and preliminaries

We consider a population of monotypic individuals that are capable of perishing and transformation into a random number of individuals of the same type. These individuals may be of biological kind, molecules in chemical reactions etc. Suppose that the population size changes according to a random reproduction law as follows. Each individual existing at the epoch t gT = [0; independently of its history and of each other for a small time interval (t; t + e) transforms into j G No\{1} individuals with probability aje + o(e) and, with probability 1 + aie + o(e) each individual survives or makes evenly one descendant (as e I 0). Where N0 = {0} U N and N = {1,2,...}. The numbers {aj} are intensities of individuals' transformation that aj ^ 0 for j G N0\{1} and 0 < a0 < -a1 = aj < to. Newly appeared individuals undergo trans-

j£no\{1}

formations under the same way as above. Letting Z(t) be the population size at the moment t, we have a homogeneous continuous-time Markov Branching Process (MBP), which was first considered by Kolmogorov and Dmitriev [13]. The process Z(t) is a Markov chain with the state space on N0. Its transition functions

Pij(t) = Pi {Z(t) = j} := P {Z(t + t)= j \Z(t) = i} , satisfy the branching property

Pij (t)= E Piji (t) • Pij2 (t) ••• Piji (t). (1.1)

ji+—+ji=j

The probabilities P1j (t) in (1.1) are calculated using the local densities {aj} by the relation

P1j(e) = dj + aje + o(e), as e I 0, (1.2)

*imomov_ azam@mail.ru © Siberian Federal University. All rights reserved

where Sjj is the Kronecker delta function. A probability Generating Functions (GF) version of the relation (1.2) is

F(e; s) = s + f (s) • e + o(e), as e I 0,

for all 0 ^ s < 1, where

F(t; s)=J2 Pj(t)sj and f (s) = ]T ajsj.

jeNo jeno

Owing to Markovian property the GF

J^Pij(t)sj = [F(t; s)]j , for all i € N. (1.3)

jeno

Assuming a := f'(1) is finite and using equation (1.4) we have ejZ(t) = ^ jPj (t) = ieat. The

jeno

last formula shows that long-term properties of MBP seem to be variously depending on the parameter a. Hence, the MBP is classified as critical if a = 0 and sub-critical or super-critical if a < 0 or a > 0 respectively. Monographs [2,5,19] are general references for mentioned and other classical facts on theory of MBP.

Throughout this paper we write P{*} and E[*] instead of Pi{*} and Ei[*j respectively. Let the random variable H := inf {t €T: Z(t) = 0} be a hitting time of the zero state of MBP, i.e. the time of extinction of the process Z(t). By the extinction theorem Pj {H < to} = qj, where q = limt^TO P10(t) is the extinction probability of MBP, which is the least non-negative root of f (s) = 0. Moreover lim F(t; s) = q uniformly for 0 ^ s ^ r < 1. Let us consider the

conditioned distribution function PH(t){*} := Pj {* |t < H < to}. It is known that if a < 0 then q =1. Therefore in this case P™^M = Pj {*\H >t} and P {t < H < to} = P {Z (t) > 0}. On the other hand in this case 0 < Pj(t) < P {H > t} ^ 0 as t ^ to. But the ratio Pj(t)/P {H > t} has a limiting finite law. So long-term properties of non-supercritical MBP are traditionally investigated on non-zero trajectories, that is under the condition of event {H > t}. Sevastyanov [18] proved that in the sub-critical case there is a limiting distribution law lim PH(t) {Z(t) = j} if

and only if aj j ln j < to. In the critical situation he also proved that if 2b := f ''(1) < to, then

jen

Z(t)/bt has a limiting exponential law. In this case Chistyakov [3] proved that if f (4)(1) < to and j/bt is bounded, then t • ph(t) {Z(t) = j} = 1/b + O (^lnt/t) as t ^ to. The author [6] restated this result without error term being on the condition of b < to only.

More interesting phenomenon arises if we observe the limit of conditioned distribution

PH(t+T){*} letting

t ^ to. In the discrete-time situation this limit represents a distribution measure, which defines a homogeneous Markov chain called the Q-process (see [2, pp. 56-60]) The Q-process was considered first by Lamperti and Ney [14]. Some properties of it were discussed by Pakes [15-17], Imomov [7,9-11], Formanov and Imomov [4]. A considerable part of the paper [12] is devoted to the discussion of this process from the viewpoint of branching transformation called the Lamperti-Ney transformation. A closer look shows that in the MBP case the limit lim PH(t+T) {Z(t) = j}

is an honest probability measure Q(t) = {Qjj(t)} which defines a homogeneous continuous-time Markov chain with state space on the set N called the Markov Q-Process (see [8]). Let W (t) be the state size at the moment t €T in the Markov Q-Process. Then W(0) = Z(0) and

Pj {W(t)= j} = Qjj(t). - 118 -

In the mentioned above paper [8] some asymptotic properties of distribution of W(t) were established. Namely, it was proved that if the corresponding MBP is critical, then W(t)/eW(t) has a limiting Erlang law. In this case there is an invariant measure if the second moment of the GF f (s) is finite. In the non-critical situation under some moment condition there exists an invariant distribution for the process W(t).

In Sec. 2 we define the Markov Q-Process and discuss properties concerning its construction and its transition function Q(t). In the Sec. 3 an ergodic property of Q(t) is established.

2. The Markov Q-process

In this section we are interested in the limiting interpretation of the conditioned transition function P^(i+T) {Z(t) = j} letting t ^ to and for all fixed t e T. First, by the law of total probability we write

Pi {t< H < to, Z(t) = j} = P {t< H < to\Z(t) = j }■ Pij(t).

Since the probability of extinction of j particles is qj, it follows that

Pi {t< H < to, Z(t) = j} = Pij (t) ■ qj. (2.1)

Using formula (3.1), from the last relation we obtain that

Pi {t< H < to} =J2 Pi {Z(t) = j, t< H < to} =J2 Pij(t)qj. (2.2)

jeN jeN

The relation (2.1) implies

Pi {Z(t) = j, t + t< H < to} = Pij(t) ■ E Pj {t< H < to, Z(t) = k} =

keN

= Pij(t) ■ E Pjk (t)qk.

ken

Therefore, considering the identity (2.2) we have

V pjk (t} qk D / \

P?<-> {Z(t) = j} = P„(t) ■ ^^pP^r ■pp07) .

^jeN p11(t+T)q Ln '

Using the ratio limit property [6, Lemma 7], after short calculation it follows that Tlim Pf(t+T) {Z(t) = j} = Pi0(t) =: Qij(t),

where ¡3 = exp{f'(q)}. It is easily seen that 0 < ¡3 < 1. Namely, ¡3 = 1 if a = 0, and ¡3 < 1 otherwise. Since F'(t; q) = ¡3t

E.^ №=ee j Pj №=m=i■

jen jen

thus, we have an honest probability measure Q(t) = {Qj(t)}. This measure defines a new stochastic process W(t), t eT, called a Markov Q-Process (MQP) is a homogeneous continuous-time Markov chain with the state space E C N (see [8]). In view of Markovian nature of this process, the transition functions Qij (t) satisfy the Kolmogorov-Chapman equations:

Qij (t + £) = J2 Qik (£)Qkj (t). (23)

ke£

Thus, the random function W(t) denotes the state size at the moment t gT in MQP, so

Qij(t) = Pi {W(t) = j} = ^Pij(t). (2.4)

Considered together, the equalities (1.2) and (2.4) entail the following important representation for the transition functions Q1j (e):

Qij(e) = Sij + pje + o(e), as e I 0, (2.5)

with the probability densities

po = 0, p1 = a1 — ln/, and pj = jqj-1aj > 0 for j G E\{1},

where {aj} are the evolution intensities of MBP Z(t). It follows from (2.5) that the GF of intensities {pj} has the form

g(s) ■.= Y,pjsj = s [f '(qs) — f '(q)] . (2.6)

je£

We see that g(1) = 0, so the infinitesimal GF g(s) completely defines the process W(t) and

o < —pi = Y pj < rx'.

je£\{i}

In the following theorem we discuss basic properties of the transition matrix Q(t) = {Qij(t)}. Herewith we will follow methods and facts from the monograph by Anderson [1].

Theorem 1. The transition matrix Q(t) of the MQP is standard. Its components Qj (t) are positive and uniformly continuous functions of t G T for all i,j G E.

Proof. According to the branching property (1.1) for the chain Z(t), we see Pij(e) = d-ij + iaj-i+1e + o(e), as e ^ 0. Hence, seeing representation (2.4)

( Qii(e) = 1 + (iai — ln/) e + o(e),

I as e 10, (2.7)

\ Qij(e)= jqj iaj-i+ie + o(e),

for all i,j G E. It follows from (2.7) that

Y,\Qj(e) — SijI = E Qij(e) + \ Qn(e) — 1| = E Qij(e) + 1 — Qii(e) < (2..1)

je£ je£\{i} je£\{i}

< 2[1 — Q^e)] —> 0, as e I 0.

So the probability measure Q(t) = {Qij(t)} is standard.

Positiveness of the functions Qij(t) is obvious owing to (2.7). Supposing e > 0, it follows from equation (2.3) that

Qij(t + e) — Qij(t) = J2 Qik(e)Qkj(t) — Qij(t) = J2 Qik(e)Qkj(t) — Qij(t) ■ [1 — Qu(e)].

ke£ ke£\{i}

The last relation gives

- [1 - Qu(e)] < -Qij(t) • [1 - Qu(e)] < Qij(t + e) - Qij(t) ^ J2 Qik(t)Qkj(e) <

ke£\{i}

< E Qkj (e) = i -Qii (e),

ke£\{i}

so \Qij(t + e) - Qij(t)| < 1 - Qii(E). Similarly

\Qij(t - e) - Qij(t)\ = \Qij(t) - Qij(t - e)\ < 1 -Qu (t - (t - e)) = 1 - Qii(E).

Therefore we obtain \Qij(t + e) - Qij(t)\ < 1 - Qii (\e\) for any e = 0 and for all i,j € E. The obtained relation implies that Qij (t) is a uniformly continuous function of t € T because lim^o Qu(e) = 1 for all i € E. □

Considering the property (1.3) it is easily seen that a GF version of (2.4) is

Gi(t; s) := EiSW(t) = £ Qij (t)sj = ß

je£

or, more obviously, that

Gi(t; s) =

where

d_ ( F(t; x) V dx \ q )

F(t; qs)^i-1

q

G(t; s), (2.8)

G(t; s) := Gi(t; s) = S °F(t X

3 dx x=qs

It is known that F (t; q) = q and F'(t; q) = 3* (see [19, pp. 52-53]). Therefore seeing (2.8) we

obtain once again that Qj (t) = Gi (t; 1) = 1.

je£

It can be easily seen that a GF version of the relation (2.5) is

G(e; s) = s + g(s) • e + o(e), as e I 0 and for all 0 ^ s < 1. (2.9)

According to formulas (1.3) and (2.8), it follows that the GF G(t; s) satisfies the following functional equation:

G(t; F(t; s))

G(t + t; s) =-^-) G(t; s), (2.10)

G (0; F(t; s))

where F(t; s) = F(t; qs)/q is the probability GF of sub-critical MBP. Using (2.9) and (2.10) for the difference AeG(t; s) = G(t - e; s) - G(t + e; s) we get

AeG(t; s) = [some function] • e + o(e), as e I 0,

for any t € T and all 0 < s < 1, which implies that G(t; s) is differentiable.

3. Classification and ergodic behavior of the transition functions {Qij(t)}

Note that evolution of MQP is ruled, in essence, by the structural parameter ¡3. Afterwards we will see that there are two types of processes depending on a value of this parameter. It has

x = qs

been shown in [8] that

G(t; s) = s exp |^ h (f(t; s}) dr^j , where h(s) = g(s)/s. Putting together this formula and (2.8) we write

G*(t; s) = s F(t; s) * 1 exp j£ h (f(t; s}) dr^j . (3.1)

Let a := g'(1) be finite. Differentiating at the point s = 1, from (3.1) we get

e*W(t) = (i - 1) 3* + eW(t)

and

( 1+ 7 (1 - 3*), when 3 < 1, EW (t) = ^ (3.2)

[ at + 1, when 3 = 1.

Moreover we obtain the variance structure

{ [7 +(i - 1)(1+ 7) 3t](1 - 3*), when 3< 1, D*W (t) = \ (3.3)

[ ait, when 3 = 1.

Where 7 = a/\ln3\ and D*W(t) = D [W(t)\W(0) = i] in (3.3). Formula (3.2) implies that when 3 = 1

e*W(t) ~ at, as t ^ to,

and if 0 < 3 < 1

E*W(t) —> 1 + 7, as t ^ to. So in the case of 3 = 1 the MQP has the transience property. Definition 1. We classify the MQP as restrictive if 3 < 1 and explosive if 3 = 1-Theorem 2. The MQP is

(i) positive if it is restrictive and a := g'(1) is finite; (ii) null if it is explosive.

Proof. To prove assertion (i) from (2.11) we get

rt N rmo) hX dx f1 hM

lo v v ' ,J Jo f(x) Jo f(x)

ln Qii(t)= f h (F(t ;0)^ dT

since F(t;0) 11 as t ^ to, where f(s) = f (qs)/q is an infinitesimal GF of a subcritical process. Herein we used the fact that lims|0 [G(t; s)/s] = Q11(t). The condition a < to implies that the integral in the right hand side converges. Hence limt^TO Q11(t) > 0. For part (ii) we recall that in this case q = 1 and h(s) = f '(s) if 3 = 1. Similarly

ln Qn(t) = f h (F(t;0)) dT = f

oo

F(t;0) h(x) ^ f1 f'(x)

■ dx —> „, s dx = — 00.

lo Jo f (x) Jo f (x)

So that limt^TO Qn(t) = 0.

Now let us recall the following statement.

Lemma 1 ( [6]). The following assertions are valid. • Let a = 0. Then

dF(t; s) \ ln[3\

ds f(s)

where

A(s) ■ ßl (1 + o(1)), as t ^œ, (3.4)

1-q

A(s) = (q — s) exp

\f'\— + frl <4 (3.5)

IJ s u — q f (u) _ J

u - q f (u)

• Let a = 0. If the second moment f ''(1) = : 2b is finite, then dF(t; s) b (1 - s)2

-r-= -2 (1 + 0(1)), as t oo. (3.6)

ds f(s) [bt(1 - s) + 1]2 ' V 7

Putting together (3.1) and (3.4)-(3.6) and considering that lim F(t; s) = 1 uniformly for all 0 ^ s ^ r < 1, we obtain the following theorem.

Theorem 3. Let a : = g'(1) is finite.

1. If MQP is restrictive, then

I ln B\

Gi(t; s) = s^A(qs)(1 + o(1)), as t ^ n, (3.7)

f (qs)

where the function A(s) has the form (3.5).

2. If MQP is explosive, then

Gi(t', S) = S 2 \ f(s)

(1 — s)

(1 — s)at + 2

2

(1 + o(1)), as t ^ œ. (3.8)

Since Q11(t) = lims|0 [G(t; s)/s], the relations (3.7) and (3.8) give the following local limit theorem.

Theorem 4. Let a := g'(1) is finite.

1. In the restrictive case

Qii(t) = MA(0) (1 + o(1)), as t ^n, a0

if only the following condition is satisfied:

-T(-^-du =< n [A]

J0 uf (q - u) A(0)

2. If MQP is explosive, then

t2Qn(t) = aa (?+T))> as t

Further on we establish limit properties of {Qj(t)} for all i,j £ E. For the general MQP the following ratio limit property holds.

Theorem 5. The limits

lim — Uj (3.9)

Qij (t)

----—"" — Uj

t^™ Qii(t)

exist for all i, j € E, and these determined by the GF

W(s) = g sj = s exp 0 ^dJj , (3.10)

where h(s) = g(s)/s and f(s) = f (qs)/q. The limiting GF W(s) converges for all 0 ^ s < 1. Proof. Let us consider the GF

Wi(t; s) = g sj = -1-Gi(t; s) = [F(t; s)] ^ W(t; s), (3.11)

j^E Q11(t) Q11(t) L J

where

w § j" ■

It follows from (3.11) that it suffices to consider the case i = 1 because F(t; s) t 1 as t ^ to uniformly for all 0 ^ s ^ r < 1. So write

W (t; s) = s exp | ^ h (FF(u; s)) - h (f(u;0)^ dJj.

One can choose r € T for any 0 ^ s < 1 so that s = FF(r;0). On the other hand we know that F (t; F(r; 0)) = F(t + r ;0) ( [19, p. 24]). Therefore we obtain equalities

W(t; s) = s exp | J h (FF(u; 0)^ du - J^ h (f(u; 0)( du^j

s exp •

o

(r r - \ M } { rh [F(t;x)) -h(x) 1

IJ \h ^(t; F(u;0))j - h (F(w;0)JJ duj — s exp j J -^-dxj>,

where f(s) := f (qs)/q. In the last step we have used the Kolmogorov backward equation

dF(t; s)

dt

— f (F(t; s)), for all 0 < s < 1,

(see [19, pp. 27-30]). To get to (3.10) it suffices to take limit as t ^ to in the obtained relation for W(t; s) being that F(t; s) ^ 1 and h(1) = 0. The assertion (3.9) follows now from the continuity theorem for a GF. Lastly it is easy to see that W(s) < to for all 0 < s < 1. □

Aggregating Theorems 4 and 5 yields the following Theorem 6. Let a := g'(1) is finite.

1. If in the restrictive case the condition [X] is satisfied, then

lln 31

Qij(t)= X(0)(1 + o(1)), as t ^ to.

ao

2. If MQP is explosive, then

(t)=aa (i+^ t ^

Now using the Kolmogorov-Chapman equation (2.3) we obtain

Qij(t + T) Qii(t + T) Qik(t) ) Qn(t + t) ' Qu(t) e Qn(t) Qkj()■

On the other hand setting s = 0 in (2.10) we see that Qn(t + t)/Qn(t) ^ 1 as t ^ x>. Hence we get the following invariance equation for [uj}:

Uj = J2 UkQkj (t), for all t GT. (3.12)

kes

The GF version of (3.12) is

W (F(t; s)) = W(s), for 0 < s < 1,

V / G(t; s)

the functional equation of generalized Schroeder form. So the set [uj} is the ergodic invariant measure for MQP.

We conclude the paper stating the following limit theorem.

Theorem 7. Let a := g'(1) is finite.

1. If MQP is restrictive, then the variable W(t) tends in mean square and with probability one to the random variable w having the finite mean and variance:

Ew =1 + y and Dw = y.

2. If MQP is explosive, then for any x > 0

lim Pi^MK < ^ e-2x — 2xe-2x■

>IW£L < xl

I EW(t) < xi

For proof see [8].

4. Conclusion remarks

The paper is devoted to the study of a population process which is defined as the long-living continuous-time Markov branching process. This is a homogeneous Markov chain called Markov Q-process (MQP). In the discrete-time situation the same process was defined in [2]. We see that the structural parameter 3 = f '(q) plays the regulating role. In fact the long-time behavior of MQP depends on this parameter and unlike the branching process it is classified only in two types. In the study of transition functions Qij (t) we essentially use asymptotic properties of the first derivative of the probability GF of Markov Branching process. As infinitesimal GF f(s) = f (qs)/q generates a sub-critical branching process, referring to [19, pp. 54-57] we see that

the condition [X] is equivalent to the convergence of the series ^j j ln j, where aj = ajqj-1.

jen

Therefore by definition of Q-process the first assertions of Theorems 4 and 6 hold under the

condition J2 Pj ln j < The Ratio limit property (Theorem 5) for transition functions states

jes

the existence of invariant measure for MQP without any moment assumptions. Theorem 7 shows the limit properties of states of process. The considered model will be extended to the age-dependent Bellman-Harris process case in our future studies.

References

[1] W.Anderson, Continuous-Time Markov Chains: An Applications-Oriented Approach, Springer, New York, 1991.

[2] K.B.Athreya, P.E.Ney, Branching processes, Springer, New York, 1972.

[3] V.P.Chistyakov, Local limit theorems in theory of branching random process, Theory of Probability and its Applications, 2(1957), no. 3, 345-363.

[4] Sh.K.Formanov, A.A.Imomov, On asymptotic properties of Q-processes, Uzbek. Matem. Zh., 3(2011), 175-183 (in Russian).

[5] T.E.Harris, Theory of Branching Processes, Die Grundlehren der Mathematischen Wissenschaften 119, Springer, Berlin, 1963.

[6] A.A.Imomov, Limit properties of transition function of continuous-time Markov Branching Processes, International Journal of Stochastic Analysis, 2014(2014).

[7] A.A.Imomov, Limit Theorem for the Joint Distribution in the Q-processes, Journal of Siberian Federal University. Mathematics and Physics, 7(3)(2014), 289-296.

[8] A.A.Imomov, On Markov analogue of Q-processes with continuous time, Theory of Probability and Mathematical Statistics, 84(2012), 57-64.

[9] A.A.Imomov, Q-processes as the Galton-Watson Branching Processes with Immigration, Trudy IX FAMET Konf., Krasnoyarsk, Russia, 2010, 148-152 (in Russian).

10] A.A.Imomov, Some asymptotical behaviors of Galton-Watson branching processes under condition of non-extinctinity of it remote future, Abst. of Com. of 8th Vilnius Conf. "Probab. Theory and Math. Statistics", Vilnius, Lithuania, 2002.

11] A.A.Imomov, On a form of condition of non-extinction of branching processes, Uzbek. Matem. Zh., 2(2001), 46-51 (in Russian).

12] F.C.Klebaner, U.Rosler, S.Sagitov, Transformations of Galton-Watson processes and linear fractional reproduction, Advances in Applied Probability, 39(2007), 1036-1053.

13] A.N.Kolmogorov, N.A.Dmitriev, Branching stochastic process, Dokl. Akad. Nauk SSSR, 56(1947), 7-10 (in Russian).

14] J.Lamperti, P.E.Ney, Conditioned branching processes and their limiting diffusions, Theory of Probability and its Applications, 13(1968), 126-137.

15] A.G.Pakes, Critical Markov branching process limit theorems allowing infinite variance, Advances in Applied Probability, 42(2010), 460-488.

[16] A.G.Pakes, Revisiting conditional limit theorems for the mortal simple branching process, Bernoulli, 5(1999), no. 6, 969-998.

[17] A.G.Pakes, Some limit theorems for the total progeny of a branching process, Advances in Applied Probability, 3(1971), 176-192.

[18] B.A.Sevastyanov, The theory of branching random processes, Uspekhi Matem. Nauk, 6(46)(1951), 47-99 (in Russian).

[19] B.A.Sevastyanov, Branching processes, Nauka, Moscow, 1971 (in Russian).

О предельной структуре марковского ветвящегося процесса с непрерывным временем

Азам А. Имомов

Институт математики Национальный университет Узбекистана Дурмон йули, 29, Ташкент Узбекистан

В статье мы исследуем предельные свойства марковского ветвящегося процесса при условии не вырождения в далеком будущем. Предельная вероятностная мера определяет случайный процесс, называемый марковский Q-процесс. Исследуем структурные и асимптотические свойства марковского Q-процесса. Изучаем асимптотические свойства переходных вероятностей и их сходимость к инвариантным ме'рам.

Ключевые слова: марковский ветвящийся процесс, марковский Q-процесс, переходные вероятности, инвариантные меры.