CC BY
12
УДК 519.218.2
On Long-time Behaviors of States of Galton-Watson Branching Processes Allowing Immigration
Azam A. Imomov*
State Testing Center under the Cabinet of MRU Karshi State University Kuchabag, 17, Karshi city, 180100
Uzbekistan
Received 13.06.2015, received in revised form 01.08.2015, accepted 02.09.2015 We observe the discrete-time Branching Process allowing Immigration. Limit properties of transition functions and their convergence to invariant measures are investigated. In the critical situation a speed of this convergence is defined.
Keywords: branching process, immigration, transition functions, invariant measures, ratio limit property, rate of convergence.
DOI: 10.17516/1997-1397-2015-8-4-394-405
Introduction
Let the random function Xn denote the successive population size of the Galton-Watson Branching Process allowing Immigration (GWPI) at the moment n G No, here N0 = {0} U {N = 1, 2,...}. The state sequence {Xn} is a homogeneous Markov chain with state space on N0 and can be expressed recursively as
Xn-1
Xn =53 £nk + Vn, for n G N,
k=i
where independent and identically distributed (i.i.d.) random variables £nk denote the offspring number of k-th individual in the (n — 1)-th generation, and i.i.d. variables nn are not depend on £nk interpreted as number of immigrants-individuals at the moment n. We assume X0 = 0 and the process starts owing to immigrants. Each individual reproduces independently of each other and according to the offspring law pk := P {£11 = k}. With probability hj := P{n1 = j} arrive j G N0 immigrants in population in each moment n G N. These individuals undergo further transformation by the reproduction law {pj}. Throughout the paper we assume p0 > 0 and
£ hj = 1.
j£No
We denote S C N0 to be the state space of the chain {Xn}. It is indicated by n-step transition functions
pj = Pi {Xn = j} := P { Xk+n = j | Xk = i} ,
for any n,k G No. Let
V^is) := Егp(n)sj
jes
*imomov_ [email protected] © Siberian Federal University. All rights reserved
is probability generating function (PGF). Denoting
one can see
From (1.1) we have
G(s) := J^ hj sj and F(s) := J^ Pjsj,
jet% jeNo
rniUs) = G(s) ■P« (F(s)). (0.1)
n — 1
rni)(s) = [Fn(s)Tl[ G (Fk(s)), (0.2)
k=0
where Fn(s) is n-fold iterate of PGF F(s); see, e.g., [1, p. 263]. Now it is clear the probabilities p(jnj are completely defined by means of probabilities {pj} and {hj}. Classification of states of the chain {Xn} is one of fundamental problems in theory of GWPI. Direct differentiation of (1.2) gives us
j = J (a-Y + ')A' - A-l • when A =!•
a
EiXn = £jj = I -A-l
jeS I an + i , when A = l,
where A = F'(l) and a = G'(l). The received formula for EjXn shows that classification of states of GWPI depends on the value of parameter A is the mean number of direct descendants of single individual as a result of transformation for one-step generation. Process {Xn} is classified as sub-critical, critical and supercritical if A < l, A = l and A> l accordingly.
The above described evolution process of individuals was considered first by Heathcote [3] in 1965. Further long-term properties of states and a problem of existence and uniqueness of invariant measures of GWPI were investigated in papers of Seneta [8,10,11], Pakes [4-7] and by many other authors. Therein some moment conditions for PGF F(s) and G(s) was required to be satisfied. In aforementioned works of Seneta the ergodic properties of {Xn} were investigated. He has proved that in cases A < l there is a unique invariant measure {pk, k 6 5} and besides ^0 = l. Heathcote [2] and Pakes [7] have shown that in supercritical case 5 is transient. In the critical case if the first moment of immigration law a := G'(l) is finite, then S can be transient, null-recurrent or ergodic. In this case, if in addition to assume that 2B := F''(l) < to, properties of S depend on value of parameter A = a/B: if A> l or A< l, then S is transient or null-recurrent accordingly. In the case when A = l, Pakes [6] and Zubkov [15] studied necessary and sufficient conditions for a null-recurrence property. Limiting distribution law for critical process {Xn} was found first by Seneta [9]. By them it has been proved under the condition of
0 < A < to the normalized process Xn/n has limiting Gamma distribution with density function
_1_ f—) A—1 e—x/B for x> 0
Br(A) \BJ e ' ^
where r(*) is Euler's Gamma function. This result without reference to Seneta has been established also by Pakes [6].
More recent researches on asymptotic properties of process contain in papers [12-14] in which the Bernoulli type GWPI was considered, i.e. both £nk and nn obey the Binomial distribution law. Clearly Bernoulli type GWPI is a special class in the general theory of Branching Processes. In this paper we consider processes in which both offspring law and immigration law are arbitrary.
In Section 2 invariant properties of GWPI will be investigated. The analogue of the Ratio Limit Property Theorem for transition functions {p(") j will be proved (Theorem 1 below).
The Section 3 is devoted to estimate of speed of convergence of j nxpj:)^ to invariant measures in the critical case.
1. Invariant property of transition functions
First we are interested in long-time behavior of ratio pjjp<00) for any i,j G S. Having designation Vn(s) := V0) (s), it follows from (1.2) that
V(i) (?)
' n (s) —> qi as n — m, (1 1)
Vn(s) q' ' 1 J
because of Fn(s) — q for 0 < s < 1; see [17, p. 53]. Recall that q is an extinction probability of the simple branching process without immigration with PGF F(s). It is the least nonnegative solution of q = F(q), and that q = 1 if A ^ 1 and q < 1 if A > 1. Putting s = 0 in (2.1) implies p(n' jp0n' — qi as n —y m. On purpose to receive the statement generally for all j G S, we write
Vn+i(s)= Vn(s) ■ G (Fn(s)) and from here and considering the properties of PGF one can calculate derivatives of j-th order:
dj Vn+i(s) dj Vn(s) G ( D ()
ds j = —j ■ G (Fn(s)) + j(s),
for all 0 < s < 1, where expression Dj,n(s) is a power series with nonnegative coefficients. Since p0n) = djVn+1(s)/dsj|s=0, from last received results we obtain
(n+1) (n)
p0j > j (n+1) ^ (n)' p00 p00
So the sequence of functions jp0n^ jp0a^ monotonously increases as n — m. In our conditions
(n)
p0o > 0 for any n G N. Therefore this sequence converges increasing to the finite non-negative limit which we will designate as Uj:
(n) p0 j
-j) t vj < m, as n — m. (1.2)
p0o
Let's consider now more general ratio pj jPw! ■ Denoting
Un (*)■■= Y, j , for 0 < s< 1,
jeS poo
we write the following equalities:
PM
UHs) = Y Pjsj = Fn(s)r = FnST Un(s), (1-3)
jeS poo
where
(n)
jes Poo
Now wee prove the following Ratio Limit Property (RLP) Theorem. Theorem 1. The general GWPI satisfies the RLP for all i,j € S:
(n)
lim Pny = qivj < (L4)
n (n)
oo
An appropriate PGF of Vj = limn^TO p0n^ jpOO is
U(s) = Y, Uj sj
jes
and it satisfies the functional equation
a ■ U(s) = G(s) ■U (F(s)), (1.5)
in a region of its convergence, where a := G(q).
Proof. The statement (2.4) immediately follows from relations (2.2), (2.3) and that fact Fn(s) ^ q uniformly for 0 ^ s ^ r < 1 as n ^ to.
To prove the justice of the equation (2.5) we consider together the relations (1.1), (2.3) and the known equality Pn+1(s) = Pn(s) ■ G (Fn(s)) and receive the following equalities:
U^Xi(s) = [Fn+i(s)]i Un+i(s) = [Fn (F(s))]i pr1ol) =
= Fn F m GFpm^=GGFnk'Uii (F
Taking limit as n ^ to from here we get to (2.5).
The Theorem is proved. □
PGF U(s) as a power series represents a continuous function in field of 0 < s < 1. According to properties of PGF it converges for all s € [0; 1 — e] and for any arbitrary small constant e > 0.
Repeatedly using the iteration of PGF F(s) in the equation (2.5) leads us to the following relation:
anU(s)= Pn(s)U (Fn(s)). (1.6)
The transition function analogue of (2.6) is
an ■ Uj = £ Vipjn). (1.7)
ies
Equality (2.7) indicates that the set of non-negative numbers {vj, j € S} represents an invariant measure for the chain {Xn}.
Due to the condition pOO > 0 and the equality (2.7), all of Vj < to and Vj > 0 for j € S. And v0 = 1 as well. Then by definition of the process {Xn} and owing to (2.7) we have the following chain of equalities:
an = an ■ vo = £ Vip(n) = ies
= £ VipiO(n)Po0) = PO^E ^wH^
ies ies
where Pi0(n) = Pi{Zn =0} is a hitting probability to zero state of the process {Zn} without immigration and generated by PGF F(s). Since this probability is equal to Fn(0), one can see an = Vn(0)U (Fn(0)) and hence
an
U (Fn(0)) = —-, (1.8)
p00
for any n G N.
Let's consider the case A = 1. Due to continuity of U(s), from equality (2.8) we receive
an
—(-) —>U(q) < m, as n — m. (1.9)
p(n)
Here we considered that Fn(0) — q. Now considering together the relations (2.1), (2.4) and (2.9), we can write the following theorem.
Theorem 2. If A = 1, then
a-npn —— JlVj—, as n — m,
ij
Z qkuk '
keS
for all i, j G S, where a = G(q) and Uj = lim pj ^ / p^ .
Further we expand our discussion concerning the equation (2.5) investigating properties of its solution.
Theorem 3. Let A = 1. Then there is a unique (up to a multiplicative constant) solution U(s) of the equation (2.5) for s G [0; q) such that
L(t) = U (q — t) (1.10)
is a slowly varying function as t ^ 0.
Proof. Let's propose that there is another solution U(s) of the equation (2.5). Then owing to equality (2.6) we write
U(s) = U (Fn(s))
Ui(s) U (Fn (s))' '
By definition the solution U(s) as well as U(s) monotonically increases. Since Fn(0) t q then for given each s G [0; q) always there is k G N such that Fk (0) < s < Fk+1(0). Hence from equality (2.11) we will receive the following relations:
U(s) < U (Fn+k+1(0)) = U (Fn+k+1 (0)) U (Fn+k+1(0)) U(sp U (Fn+k(0)) U (Fn+k+1 (0))' U (Fn+k(0)) '
But again according to equality (2.11)
U (Fn(0)) = U(0) = 1 U (Fn(0)) 12(0) '
Then using once again (2.6) and the formula Vn+1(s) = Vn(s) ■ G (Fn(s)), we have
U(s) < U (Fn+k+1 (0)) = a
U(s) U (Fn+k(0)) G (Fn+k(0))
Taking limit as
Us) < 1
U(s)
because PGF G(s) continuously. By the similar way it is possible to establish the converse inequality U(s) jU(s) > 1. The received conclusions say that the equation (2.5) has a unique solution for all s € [0; q).
Now following a method of Seneta [8] we put
g(s) = G(q — s), f (s) = q — F (q — s).
In the allowed designations the equation (2.5) becomes
a ■L(s)= g(s) ■L (f (s)), for s € [0; q). (1.12)
We will be convinced the function f (s)/s monotonically decreases on set of 0 < s < q taking here the maximum ¡3 = lims|0 [f (s)/s] and the minimum f (q)/q = 1 — p0/q accordingly, where as before 3 = F'(q). The function L(s) in form of (2.10) is also monotonically decreasing on this set and lims|0 g(s) = a. Then for any A € [3; 1] one gets
c(fs) s)
L (f(s))= { s J > OM > -A) > L
L(s) L(s) ' L(s) L(s)
On the other hand according to (2.12),
L (f (s)) a 1
rt , = —TT —> 1 as s I L(s) g(s)
Hence,
L(As) lim = 1,
s|o L(s)
for any A € [3; 1]. It is easy to be convinced that the last relation is valid for any A € R+, where R+ is set of positive real numbers. So L(s) = U(q — s) is a slowly varying function as s I 0. The Theorem is proved. □
In the critical case it has been proved by Pakes [5] that the sequence {nxPn(s)} converges to the limiting PGF n(s) uniformly for 0 ^ s ^ r < 1 which is a solution of the equation (2.5):
n(s) = G(s)n (F(s)),
where as before A = a/B. It was supposed therein that the moments
£pjj2 ln j and E hjj ln j
jes jes
are finite. An advantage of assertion of the Theorem 1 from aforementioned result of Pakes consists that in our case the invariant measure {Vj} for GWPI and corresponding for it the equation (2.5) is established without any moment assumptions concerning distributions {pj} and {hj}. In the final Section of the paper we investigate a speed of convergence
nxPn(s) —> n(s), as n ^ to, strengthening aforementioned result of Pakes.
2. A speed rate of convergence to invariant measures in critical situation
Consider the critical GWPI with transition functions pj = Pi {Xn = j}. Recall the appropriate PGF
n-1
Vn(s) = J2 pj ]s° = n G (Fk (s)), (2.1)
jeS k=0
where G(s) and F(s) are PGF's of immigration stream law and the process offspring law accordingly. Provided that moments G''(1) and FIV (1) are finite, Pakes [5] investigated a rate
of convergence {nxVn(s)} to the limiting PGF n(s) = ^ njsj, which is being the solution of
jeS
functional equation n(s) = G(s) ■ n (F(s)). So the nonnegative numbers {nj} satisfy the relation
E(n)
nHpij .
ieS
In this section we improve aforementioned result of Pakes, holding to condition of the third order factorial moment of offspring PGF is finite.
Theorem 4. Let A =1, 2B := F''(1), a := G'(1) and X = a/B. If C := F'''(1) < m, then the
sequence {nAPn(s)} converges to n(s) uniformly for 0 ^ s ^ r < 1, and besides
n bn(s
bn(s)
nxVn(s) = n(s) ■ (l + A ■ (1 + o(1))^j , as n ^ œ, (2.2)
where A := a f 6BB2 — ^ and
0
bn(s) = Bn + 1
1 — s
Proof. It follows from (3.1) that
n-1
Pn(s) = n*n G (Fk(s)) =
k=0
n-1 / -, \ A n-1
= G(s)U[l + - G (Fk (s)) = G(s)n Ak (s),
(2.3)
where Ak (s) = ( 1 + 1 J G (Fk (s)). It is known that the infinite product H Ak(s) and the
V k/ keN
series (Ak(s) — 1) converge or diverge simultaneously. Therefore we investigate the last series.
keN
Using elementary expansion (1 + 1/k)A = 1 + X/k + £k, we have the following representation:
Ak(s) — 1 = X — (1 — G (Fk(s))) — X (1 — G (Fk (s))) + £kG (Fk (s)), (2.4)
where £k = O {1/k2). We write
1 — G(s) = a ■ (1 — s) — S(s)(1 — s), (2.5)
where 0 < S(s) = (1 — s) G''(0)/2 and s < 0 < 1; obviously that S(s) = o(1) as s t 1. In turn we know that (see [1, p.74]):
\1 — Fn(s) \ < 2(1 — Fn(0)), (2.6)
and according to the Basic Lemma of the theory of critical processes 1 — Fn(0) ~ 1/Bn; see e.g. [1, p.19]. Hence from relations (3.4)-(3.6) we will easily be convinced that
Ak(s) — 1 = O ) ' as k ^ to,
uniformly for 0 ^ s ^ r < 1. Last statement testifies to a uniform convergence of the series
Y^ (Ak(s) — 1) and hence the infinite product G(s) Ak(s). We denote this as
keN keN
n(s) ■= lim nxPn(s) = G(s) n Ak(s). (2.7)
keN
Now to the proof of relation (3.2), we will estimate the error term of difference nxPn(s) — n(s). So using representation (3.3) and equality (3.7) we obtain
nxPn(s) — n(s) = G(s)
n — 1 CO
]^[Ak (s) — n Ak(s)
k=1 k=1
(2.8)
1
G(s)H Ak (s)
k=1
1 — n Ak (s)
According to the positiveness property of PGF we see Ak (s) > 0 for all 0 < s < 1. Then using the elementary inequality ln (1 — x) > —x — x2/(1 — x) we write down the following equalities:
ln]J Ak (s) = J2ln {1 — (1 — Ak (s))}
n(1)(s) = ■ Y.(,)+ p(1)
k(
k^n k^n
— £ (1 — Ak(s)) + pW(s) =■ Xn(s) + p[1)(s),
(2.9)
k^n
where
and
Xn(s) = — E (1 — Ak(s)),
k^n
0 > n)(s) > -T [1 — Ak (s)]2 > 0 > ^ (s) > fe Ak(s)
> — (1 — G (Fn(s))) £ ^ >
k^n ky '
> 1 — G (Fn(s)) y ( )
> G (Fn(s)) ^ hn(s)-
The monotone property of PGFs used in the last step. Replacing s by Fk(s) it follows from (3.5) that
1 — G (Fn(s)) = a (1 — Fn(s)) + S (Fn(s)) (1 — Fn(s)). (2.10)
k
Owing to (3.6) and (3.10) 1 — G (Fn(s)) ~ A/n and hence the augend in (3.9) Pn\s) ^ 0 always supposing the first term £n(s) has a finite limit as n ^ to. In turn in our conditions and owing to (3.6) 5 (Fn(s)) = O (1/n). Therefore combining (3.4), (3.6) and (3.10) we will receive the following equality for first term in (3.9):
Sn(s) = — £ (a (1 — Fk(s)) — A) + E O (T) • (2.11)
Further we use the following asymptotic expansion for the function 1 — Fn(s) which holds in the conditions of our theorem:
1 Pi) 1 . A InbuS+Kis) (212)
1 — Fn(s) = ^r^+A-----(1 + o(1)), (2.12)
bn(s) (bn(s)r
~ C
as n ^ to, where A = —- — B and K(s) is some bounded function depending on form of
6B
F(s) and bn(s) is same as in the theorem statement. The formula (3.12) was established in the paper [16] and we have reduced it in a bit modified form. So bn(s) = O(n) as n ^ to, considering the expansion (3.12), we rewrite the equality (3.11) in form of
Zn(s) = —aA x(k)+ f™(s), (2.13)
k^n
where fJn\s) = Y^ O (1/k2) and
k^n
xk = WTW ■
(bk (s))
One can see that the function x(k) is positive and monotonically decreases with respect k € N and for all 0 < s < 1.
We consider now an alternative function x(t) for t € R+. Obviously this function is positive, monotonically decreases and also is continuous. Moreover
J x(t)dt = X(t) + const,
where
= 1 (^M*) + 1 \
for 0 < s < 1 and X(t) ^ 0 as t ^ to. Therefore due to the Mc'Loren-Cauchy test (see [18, pp. 283-284]) the following inequalities hold:
1 lnbn(s) + _P < V- x(k) < 1 lnbn-i(s) + 1 B bn(s) bn(sp ¿p 1 ^B bn-i(s) bn-i(s)'
By means of the last inequalities and considering that bn(s) = O(n) as n ^ to we write estimation
1 ln bn (s)
x(k) —
k> B bn(s)
k^n
O ^nj , as n ^ to, (2.14)
for 0 < s < 1.
Almost obviously that pn (s) = ^ O (1/k2) = O (1/n). Considering it and exploiting (3.14)
tf) (s) =
k^n
in (3.13) we will obtain
Vn(s) = —A ■ m "n\a; + o(, as n ^to, (2.15)
n
ln bn(s) f 1
Vn(s) = —A ■
bn(s) \ny
where A = ^ 6B2 — 1).
Return now to the equality (3.9). Due to (3.15) and the fact 1 — G (Fn(s)) ~ A/n arises Pn\s) = O (lnn/n2). Then from (3.9) and (3.15) we conclude
]jAk (s)=exp{ —A ■ iniss)(1 + 0 W)} ' as n ^ to.
k^n
Finally using last expansion in (3.8) with combination of the formula 1 — e—x ~ x, x ^ 0, we complete the theorem proof. □
From Theorem 4 we receive the following Corollary 1. In conditions of Theorem 4 the following assertion is valid \ (ri) ( A ln n f\n n\\
nXPoo] = no{1+ B ■ — + 0{—))> as n ^TO'
where A is defined in Theorem 4.
The following theorem generalizes the previous one.
Theorem 5. Let conditions of the Theorem 4 are satisfied. Then the sequence j^Pn^s) j converges to the limiting function n(s) uniformly on the set of 0 ^ s ^ r < 1, and
nPHs) = n(s^Sni)(s) + A(ni)(s) ■ ^fiss- (1 + 0 (1)))
(1)) , (2.16)
bn(s)
as n ^ to, where An \s) = A ■ Sni)(s) and
Sni)(s) = 1 — rh'
bn(s)
and expressions A and bn(s) are defined in Theorem 4.
Proof. Since Fn(s) < 1 and Fn(s) t 1 as n ^ to, it follows from (1.2) and Theorem 4 that jnP^s)} converges uniformly to n(s) for 0 ^ s ^ r < 1. Write
nAPni)(s) = (Fn(s))i nxPn(s). (2.17)
It is obvious for fixed i and at large values of number n
(Fn(s))i = 1 — i (1 — Fn(s))(1 + o(1)). From here and using (3.12) follows
(Fn(s))i = 1 — T-^ (1 + 0(1)), as n ^ to. (2.18)
bn(s)
Now the theorem statement follows from equalities (3.17) and (3.18), with application of the
statements (3.2) and (3.12). □
Corollary 2. In conditions of Theorem 5 the following assertion is valid:
nXp(n) = 5W + A5P ■ Bn (1 + °(1))) , as n ^ to,
for all i,j € S, where 5^ = 1 — i/Bn.
References
[1] K.B.Athreya, P.E.Ney, Branching processes, Springer, New York, 1972.
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Об асимптотическом поведении состояний ветвящихся процессов Гальтона-Ватсона с иммиграцией
Азам А. Имомов
В 'работе рассматривается ветвящийся процесс с иммиграцией дискретного времени. Исследуются предельные свойства переходных вероятностей и их сходимость к инвариантным метрам.
В критическом случае определяется скорость этой сходимости.
Ключевые слова: ветвящийся процесс, иммиграция, переходные вероятности, инвариантные меры, скорость сходимости к инвариантным метрам.
