CC BY
14
Journal of Siberian Federal University. Mathematics & Physics 2014, 7(3), 289-296
УДК 519.21
Limit Theorem for the Joint Distribution in the Q-processes
Azam A. Imomov*
Karshi State University, Kuchabag, 17, Karshi, 180100, Uzbekistan
Received 10.02.2014, received in revised form 10.03.2014, accepted 23.04.2014 The Q-process is considered in this paper. A link between the Q-process and the Galton-Watson branching process allowing immigration is established in the paper. Due to this link the limit theorem on the joint distribution of the population size and the total state of the Q-process is proved.
Keywords: Galton-Watson branching process; immigration; total state of the Q-process; limit theorem.
1. Preliminaries
We consider the so-called Q-processes. The Q-process is defined by the Galton-Watson process (GWP) conditioned on non-extinction of its trajectory in the distant future. We show that the Q-process may be replaced by some GWP allowing immigration (GWPI). We investigate asymptotic properties of the joint distribution of the population size and the total state in Q-processes.
1.1. On simple GWP. Let us consider a GWP. Let Zn, n e N0 (N0 = {0} U {N =1,2,...}), be the number of individuals in the nth generation defined recursively as
Zn-1
Z0 — 1j Zn — ^^ Cn
k=1
where independent and identically distributed (i.i.d.) random variables Zn,k denote the offspring of k-th individual in the (n — 1)th generation. Let P {Zi = k e N0} =: pk be an offspring law of the single individual and p0 > 0, p0 + p1 = 1. According to the branching condition the evolution law of GWP is regulated by the probability generating function (GF) F(s) := pksk,
fceNo
|s| < 1 and Fn(s) := EsZn is determined by the n-step iteration of F(s). In this interpretation A := F'(1) = EZnk is the mean per capita number of offsprings [see, e.g., 1, pp. 1-2].
We know that when A < 1 and A =1 the GWP is die out asymptotically. Accordingly, in these cases the properties of GWP are investigated on nonzero trajectories. In this context we
n— 1
recall the following theorem on joint distribution of Zn and Yn := Zk. The variable Yn
k=0
denotes the total number of offsprings of single individual until time n in GWP.
Theorem 1.1 (see, e.g., [11, p. 143]). Let A = 1 and F''(1) < to. Then two-dimensional random vector (Zn/E [Zn| Zn > 0]; Yn/E [Yn| Zn > 0]) weakly converges to the unique vector (Z*; Y*) as n ^ to and the Laplace transform for (Z*; Y*) is of the form
E
-\Z*~9Y*
sh2\fd 2^0
+ A
'imomov [email protected]
© Siberian Federal University. All rights reserved
shv^
"TT,
k
1
2
The limit Laplace transform has been also obtained by D.Kennedy [10] in the study of the behaviour of Zn conditioned on the event {Y := lim Ym = n} as n ^ ro. Setting A = 0 gives
m—
the well-known conditional limit law for 2Yn/F''(1)n derived by A. Pakes [13, Theorem 4].
This theorem also contains (at d = 0) the well-known A.Yaglom's result [19]. According to that result, limit of conditional distribution
( 2Z
Sn,m(x) := P | f''(1)n ^ X l Z"+m > 0
exists as n ^ ro for any m G No, and lim Sn m(x) = 1 — e-x, x > 0.
n—'
1.2. On Q-processes and main result. It was shown that the use of the conditioning of not-extinction in the distant future > 0 instead of fundamental conditioning Zn > 0 gives a new limit distribution law which is distinct from classical one (conditioning of Zn under > 0 means P {Zn |ZTO > 0 } = lim P {Zn |Zn+m > 0 }). T. Harris [3] pointed out that at the same
m—
conditions as in Theorem 1.1 the following result holds:
lim lim Sn m(x) = 1 — e-x — xe-x, x > 0.
n—m—'
Later the condition > 0 was treated by J.Lamperti and P. Ney [12], K.Athreya and P. Ney [1, pp. 56-60], A. Pakes [13,15,17], A. Imomov [5,6,8], Sh.Formanov and A. Imomov [2]. Continuous time case was discussed in [7,9,18].
The stochastic process {Wn, n G N0} defined by GWP under conditioning of > 0 is called the Q-process in [1, pp. 56]. In fact the Q-process {Wn, n G N0} is homogeneous Markov chain with zero state W0 = 1 and it is given by transition probabilities
Qjn) = P { Wn+k = j | Wfc = i} =
= P { Zn+fc = j | Zfc = i, > 0} ,
for n, i, j, k G N. As it was proved in [1, pp. 56-58] these probabilities are of the form
Qjn) = p {Zn+k = j | Zk = i}, (1.1)
where q is an extinction probability of GWP for which q = F(q) and ft := F'(q).
Further we need the GF W^s) := E Q(n) sj. Taking into account the branching property
of GWP and (1.1), we have the following relation:
Wii)(s) =
Fn(qs)
q
Wn(s), (1.2)
where the GF Wn(s) := Wni)(s) = E |W0 = ^ i
is
F'(qs)
Wn(s) = s^ft^ , n G N. (1.3)
We see that the Q-process {Wn, n G N0} can be defined by GF W(s) = W1(s). Assume that a := W'(1) has a finite value. We find out that a = 1 + qF''(q)/ft > 1.
We study a limit of a joint distribution of the Q-process and its total state for the case A =1 (critical case for GWP). By the total state in the Q-process we mean the variable
Sn := W0 + Wi + • • • + Wn-1, S0 =0.
Theorem 1.2. Let A = 1. Then the random vector (W„/EW„ ; Sn/ESn) weakly converges to a random vector (W*; S*) as n ^ to. The vector has the following Laplace transform
-2
, A,0 > 0.
We note that the same limit Laplace transform appears in paper of A. Pakes [17] but in the different context.
Further we use the joint GF
Jn(s; x) := Y^ {Wn = i, Sn = k} sixk, (s; x) G D, (1.4)
¿eNo fceNo
on the set D ={ |s| < 1, |x| < 1 : ^(s - 1)2 + (x - 1)2 > r> oj.
In Section 2 we establish a link between the Q-processes and GWPI. This link allows us to find out the necessary relations for GF Jn(s; x). In Section 3 we discuss several preliminary results on properties of GFs when A =1. These results are used to prove the Theorem 1.2 in Section 4.
E
-xw *-es*
cw*+2 mp
2 yß
e
2. Q-processes as GWPI
By iterating F (s), the GF (1.3) may be written as
Wn(s) = -nV^), (2.1)
k=0
with G(s) = F'(qs)/^. It is easy to see that the following random sum of random variables is comparable with GF (2.1):
Wn-1
Wo = 1, W„+i = 1 + ^ £n+i, k + nn+i, (2.2)
k=1
where ^n,k are i.i.d. random variables with common GF F (qs)/q for all n and k. Variables nn are i.i.d. random variables with Esnn = G(s). Then we can conclude that the Q-process may be replaced by the following branching process. In the beginning there is one particle. The evolution process is initiated by the stream of the immigrating particles. The emergence intensity law is described by GF G(s). The immigrating particles in prospect undergo a transformation according to the GF F (qs)/q. In addition, the initial particle does not disappear and does not breed. This "immortal particle" is present throughout the evolution of the process. Upon introducing Wn = Wn — 1, relation (2.2) is written in the form
W n-1
W n = E
£n,k + nn. (2.2*)
k=1
One can see that the sequence {Wn, n G N0} is nothing but the GWPI with W0 = 0 and
transition probabilities Qj := P {Wn+k = j |W k = ¿} = j+1. Here Z „+i = £n,k is
k=1
the "internal" GWP that obeys the GF Es^nfc = F (qs)/q and arrival intensity of immigranting
particles is regulated by GF G(s). We refer the reader to C.Heatcote [4] and A. Pakes [14,16] on further details regarding the GWPI with general GF G(s).
_(i)
It is obvious that relations similar to relations (1.2) take place for the GF Wn (s) :=
J2 Qij1^. Thus the analysis of asymptotic properties of Q-processes may be reduced to studying jeN _
of corresponding properties of GWPI. Moreover the GWPI {Wn, n G No} may be not supercritical. Indeed, the mean number of the single individual offspring in "internal process" is
E£„fc — -d
ds
F (qs)
1 , A — 1, в < 1, A — 1.
(2.3)
n-1
The variable Yn := ^ Zk denotes the total progeny in the process {Zn, n G N0} up to
fc=0
time n. Let us define a joint GF Hn(s; x) := EsZnxYn, (s; x) G D. Following the reasoning given in [13], the following relations hold for the GF Hn(s;x):
Ho(s; ж) Hn+i(s; ж)
F (qH„(s; ж))
(2.4)
n- 1
The variable
E Wk represents the total progeny in the GWPI, defined by rela-
fc=0
tion (2.2*). We see that the joint GF Jn(s; ж) :— EsWn, (s; ж) G D has the form
n-1
Jn(s; x) — ft G (Hk(s; ж)),
(2.5)
fc=0
where GF G(s) is given in (2.1) and Hn(s; x) satisfies equations (2.4) (see also [17]).
Considering (2.2) and (2.2*) we see that Sn = Sn — n and, therefore, Jn(s;x) = sxn Jn(s;x). Now using (2.4) and (2.5), we obtain the following representation for GF Jn(s; x) defined by (1.4):
n1
Jn(s; ж) — s JJ
fc=0
жF' (qfffc(s; ж))
в
(2.6)
3. Some discussion on generating functions in the case A = 1
From now on we consider the case A = 1. In this case relation (2.6) becomes
1
Jn(s; ж) — s JJ [sF' (Hfc(s; ж))],
(3.1)
fc=0
where Hn(s; x) = EsZnxYn. We have dJ„(s; x)
ds
EWn
&
dJn(s; ж)
(s;x)=(1; 1)
дж
ESn
(s;x) = (1;1)
Using direct differentiation, it can be found from (3.1) that
EWn
(a - 1)n +1 &
ESn
a — 1
n(n + 1) + n,
(3.2)
q
1
s
s
q
where, as before, a = W'(1) = 1 + F''(1).
Further we use the GF A„(s; x) := h(x) — Hn(s; x), n G No, where h(x) = ExY is the GF of Y = lim Yn. This variable denotes the total number of particles participating in process
n—
{Zn, n G N0} for the duration of its evolution. Since the process dies out with probability 1 in the discussed case, the variable Y always exists.
By virtue of P{Zn > 0} = O (1/n) as n — to (see [1, p.11]) we have
Sup |An(s; x)| —^ 0, n — to. (3.3)
(s;x)£B
Using arguments given in [11, p. 127, Lemma 3], we obtain
|Afc(s; x)| < |Aj(s; x)|, (3.4)
for all k G N0 and j = 0,1,..., k.
We also know that for An(s; x) the following asymptotic expansion holds:
1 1
An(s; x) un(x)
u(x) := xF' (h(x)), b(x) :=
1 b(x) [1 - un(x)] ^ .
+ -rT^ + / , £fc (s; x)ufc(x)
— it ■T ) i—'
k = 1 where
Ao(s; x) 1 — u(x)
F" (h(x)) 2F' (h(x)) '
(3.5)
and Sup |en(s;x)| < en ^ 0, n ^to (see [11, p. 136].
(s;x)£D
Further we consider the behavior of h(x) and u(x) in a neighborhood of x =1. It is known [11, p. 126] that
1 — h(x) - y2(1 — x)/F''(1), x ^ 1. (3.6)
On the other hand, we have
F'(h(x)) - 1 — F''(1)(1 — h(x)), x ^ 1 (3.7)
by means of Taylor expansion. Considering relations (3.6) and (3.7) together, we obtain
u(x) - 1 — V2F''(1)(1 — x), x ^ 1. (3.8)
4. Proof of the Theorem 1.2
We follow the method proposed by A. Pakes [14]. Let ^n(A; 0), A, 0 > 0, be the Laplace transform of variable (Wn/EWn ; Sn/ESn). Taking into consideration (3.1) and (3.2), we obtain
n— 1
*n(A; 0) - Jn(An; 0n) = AnC II F' (H(An; 0n)), n ^ to, (4.1)
k=0
where An = exp{ —A/(a — 1)n}, 0n = exp{—20/(a — 1)n2}. We see that the term An0JJ on the right-hand side of (4.1) tends to unity as n ^ to. It is ensured that Akn(A; 0) := F' (Hk(An; 0n)) does not decrease with k for a fixed n and A > 0, 0 > 0. Then using the inequality ln(1 — x) >
—x — x2/(1 — x), 0 ^ x < 1, we obtain
n—1 n—1
lnf! Akn(A; 0) = ]Tln{1 — (1 — Akn(A; 0))}
fc=0 fc=0
n1
(1 - Afcn(A; 0)) + рП1)(А;|
fc=0
where
and
— 1n(A; 0) + p^(A; 0), (4.2)
n-1
1n(A; 0) — (1 - Afcn(A; 0)), (4.3)
fc=0
1
П^ ^ \ " [1 - Afcn(A; 0)] .
> 0). (4.4)
It is easy to see that 1 — A0n(A; 0) = 1 — F'(An) ^ 0 as n ^ ro. Then pn^A; 0) ^ 0 if 1n(A; 0) has a finite limit as n ^ ro. By Taylor expansion we have
F '(t) = F '(t0) — F "(i0)(t0 — t) + (t0 — t)g(t0; t), (4.5)
where g(t0; t) = (t0 — t)F'''(t)/2 with t < t < t0. Setting t = Hk(An; 0n) and t0 = h(0n) in (4.5) and using the GF Ak(s; x), equation (4.3) becomes
n1
n—1 n—1
In (A; 0) = — [1 — F '(h(0n))] n — F "(h(0n))£ Ak (An; 0n) + pn2) (A; 0), (4.6)
k=0
where
0 < pn2)(A; 0) = Ak (An; 0n )gkn(A; 0) < A0 (An; 0n) gkn(A; 0). (4.7)
k=0 k=0 Here we use inequality (3.4) and relation gkn(A; 0) := g (h(0n); Hk(An; 0n)). Using relation (3.6) we get
A0(An; 0n) - ^/'(ijn, n ^ro. (4.8)
Considering (3.3), we see that gkn(A; 0) ^ 0 as k ^ ro for all n G N0. Hence the arithmetical
1 n—1
mean of these expressions — ^ gkn(A; 0) ^ 0 as n ^ ro. Then it follows from (4.7) and (4.8)
n k=0
(2)
that pn (A; 0) approaches zero as n ^ ro. Therefore, it follows from (3.6), (3.7) and (4.6) that
n—1
In (A; 0) = — 2v^ — F''(1^Ak(An; 0n) + o(1), n ^ ro. (4.9)
k=0
Consider now the sum on the right-hand side of (4.9). From (3.5) we have
Ak(An; 0n)=—1—b(e :;(0i )1-, (4.10)
1__I b(en)l1—"k(en)] I v i\-0'\
Ac(A„;e„) + 1—«(e„) +^kn(A; 0)
where Sfc„(A; 0) := ^ £fc(A n; 0n)u (0n ). Since |efc(A„; 0„)| < e^ ^ 0, k ^ to, and |u(x
k=1
|xF' (h(x))| < F'(1) = 1, then
-Sfc„(A; 0) = o(1), n ^ to. n
Using relations (3.6)-(3.8), it is easy to find that as n ^ to
1 - U(0n) = — (1 + 0(1)) ,
n
F ''(1)
b(0n) = -^(1 + 0(1)) .
Taking into account equations (4.8), (4.10)-(4.13) we see that
(4.11)
(4.12)
(4.13)
Afc(A„;0„)—
4^
Ue
F"(1)» (l — e"2'^/») ,, +1
n2
as n ^ to, where ^ = ^A — 2%/0^ j4%/0. Then the second term in expression (4.9) can be transformed to
1
1
F"(1)5>fc(A„; ë„) = ]T
(An;en '=0 '=0
=0 fl — e-2fc^M u +1 »
!+ o(1).
The sum on the right-hand side of the last equation can be recognized as the upper (if ^ +1 < 0) or the lower (if ^ +1 > 0) Darboux sum of the Riemann integral
4u%/ëi
1 — e-2Vëx) u + 1
-dx = 2ln
1 — e-2^') u + 1
Then we finally obtain
1n(A; ë) = — 2\fë — 2ln
A — 2f f 1 — e^) + 1
We v ;
+ o(1), n ^ TO.
(4.14)
After considering relations (4.1), (4.2) and (4.14), we complete the proof of the theorem.
Acknowledgement. The author is grateful to the anonymous referee for careful reading and for contributions to improve the paper.
1
References
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Одна предельная теорема для совместных распределений в Q-процессах
Азам А. Имомов
Мы рассматриваем Q-процессы. Устанавливается глубокий связь между Q-процессами и ветвящимися процессами Гальтона-Ватсона с иммиграцией. Доказывается предельная теорема для совместных распределений состояний и общих состояний в Q-процессе.
Ключевые слова: ветвящееся процессы Гальтона-Ватсона; иммиграция; Q-процессы; общее состояние Q-процесса; предельная теорема.
