On application of slowly varying functions with remainder in the theory of Markov branching processes with mean one and infinite variance
UDC 519.218.2 We investigate an application of slowly varying functions (in sense of Karamata) in the theory of Markov branching processes. We treat the critical case so that the infinitesimal generating function of the process has the infinite second moment, but it regularly varies with the remaind...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507083931123712 |
|---|---|
| author | Imomov, A. Meyliyev , A. Imomov, Azam Imomov, A. Meyliyev , A. |
| author_facet | Imomov, A. Meyliyev , A. Imomov, Azam Imomov, A. Meyliyev , A. |
| author_sort | Imomov, A. |
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| datestamp_date | 2025-03-31T08:47:35Z |
| description | UDC 519.218.2
We investigate an application of slowly varying functions (in sense of Karamata) in the theory of Markov branching processes. We treat the critical case so that the infinitesimal generating function of the process has the infinite second moment, but it regularly varies with the remainder. We improve the basic lemma of the theory of critical Markov branching processes and refine known limit results. |
| doi_str_mv | 10.37863/umzh.v73i8.684 |
| first_indexed | 2026-03-24T02:03:41Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v73i8.684
UDC 519.218.2
A. Imomov, A. Meyliyev (Karshi State Univ., Uzbekistan)
ON APPLICATION OF SLOWLY VARYING FUNCTIONS WITH REMAINDER
IN THE THEORY OF MARKOV BRANCHING PROCESSES WITH MEAN ONE
AND INFINITE VARIANCE
ПРО ЗАСТОСУВАННЯ ПОВIЛЬНО ЗМIННИХ ФУНКЦIЙ IЗ ЗАЛИШКОМ
У ТЕОРIЇ МАРКОВСЬКИХ РОЗГАЛУЖЕНИХ ПРОЦЕСIВ З ОДИНИЧНИМ
МАТЕМАТИЧНИМ ОЧIКУВАННЯМ ТА НЕСКIНЧЕННОЮ ДИСПЕРСIЄЮ
We investigate an application of slowly varying functions (in sense of Karamata) in the theory of Markov branching
processes. We treat the critical case so that the infinitesimal generating function of the process has the infinite second
moment, but it regularly varies with the remainder. We improve the basic lemma of the theory of critical Markov branching
processes and refine known limit results.
Дослiджується застосування повiльно змiнних функцiй (у сенсi Карамати) в теорiї марковських розгалужених
процесiв. Критичний випадок трактується так, що iнфiнiтезимальна генеруюча функцiя процесу має нескiнченний
другий момент, але регулярно змiнюється з залишком. Покращено основну лему теорiї критичних марковських
розгалужених процесiв та уточнено вiдомi граничнi результати.
1. Introduction and main results. 1.1. Preliminaries. We consider the Markov branching process
(MBP) to be the homogeneous continuous-time Markov process \{ Z(t), t \geq 0\} with the state space
\scrS 0 = \{ 0\} \cup \scrS , where \scrS \subset \BbbN and \BbbN = \{ 1, 2, . . .\} . The transition probabilities of the process
Pij(t) := \BbbP \{ Z(t) = j | Z(0) = i\}
satisfy the following branching property:
Pij(t) = P i\ast
1j (t) for all i, j \in \scrS , (1.1)
where the asterisk denotes convolution. Here transition probabilities P1j(t) are expressed by relation
P1j(\varepsilon ) = \delta 1j + aj\varepsilon + o(\varepsilon ) as \varepsilon \downarrow 0, (1.2)
where \delta ij is Kronecker’s delta function and \{ aj\} are intensities of individuals transformation such
that aj \geq 0 for j \in \scrS 0\setminus \{ 1\} and
0 < a0 < - a1 =
\sum
j\in \scrS 0\setminus \{ 1\}
aj < \infty .
The MBP was defined first by Kolmogorov and Dmitriev [8] (for more detailed information see [2]
(Ch. III) and [5] (Ch. V)).
Defining the generating function (GF) F (t; s) =
\sum
j\in \scrS 0
P1j(t)s
j it follows from (1.1) and (1.2)
that the process \{ Z(t)\} is determined by the infinitesimal GF f(s) =
\sum
j\in \scrS 0
ajs
j for s \in [0, 1).
Moreover, it follows from (1.2) that GF F (t; s) is unique solution of the backward Kolmogorov
equation \partial F/\partial t = f(F ) with the boundary condition F (0; s) = s (see [2, p. 106]). If m :=
:=
\sum
j\in \scrS
jaj = f \prime (1 - ) is finite, then F (t; 1) = 1 and due to Kolmogorov equation it can be
c\bigcirc A. IMOMOV, A. MEYLIYEV, 2021
1056 ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 8
ON APPLICATION OF SLOWLY VARYING FUNCTIONS WITH REMAINDER . . . 1057
calculated that \BbbE [Z(t)| Z(0) = i] =
\sum
j\in \scrS
jPij(t) = iemt. Last formula shows that long-term
properties of MBP are various depending on value of parameter m. Hence, the MBP is classified as
critical if m = 0 and sub-critical or supercritical if m < 0 or m > 0, respectively. Monographs
[1 – 3, 5] are general references for mentioned and other classical facts on theory of MBP.
In the paper we consider the critical case. Let R(t; s) = 1 - F (t; s) and
q(t) := R(t; 0) = \BbbP \{ \scrH > t\} ,
where the variable \scrH = \mathrm{i}\mathrm{n}\mathrm{f}\{ t : Z(t) = 0\} denotes an extinction time of MBP. Then q(t) is the
survival probability of the process. Sevastyanov [11] proved that if f \prime \prime \prime (1 - ) < \infty , then the following
asymptotic representation holds:
1
R(t; s)
- 1
1 - s
=
f \prime \prime (1 - )
2
t+\scrO (\mathrm{l}\mathrm{n} t) as t \rightarrow \infty (1.3)
for all s \in [0, 1) (see [11, p. 72]).
Later on Zolotarev [12] has found a principally new result on asymptotic representation of q(t)
without the assumption of f \prime \prime (1 - ) < \infty . Namely, providing that g(x) = f(1 - x) is a regularly
varying function at zero that is
\mathrm{l}\mathrm{i}\mathrm{m}
x\downarrow 0
xg\prime (x)
g(x)
= \gamma
with index 1 < \gamma = 1 + \alpha \leq 2, he has proved that
q(t)
f(1 - q(t))
\sim \alpha t as t \rightarrow \infty . (1.4)
Further, we assume that the infinitesimal GF f(s) has the following representation:
f(s) = (1 - s)1+\nu \scrL
\biggl(
1
1 - s
\biggr)
(1.5)
for all s \in [0, 1), where 0 < \nu < 1 and \scrL (x) is slowly varying (SV) function at infinity (in sense of
Karamata, see [10]).
Pakes [9], in connection with the proof of limit theorems has established, that if the condition
(1.5) holds, then
1
R(t; s)
= U
\biggl(
t+ V
\biggl(
1
1 - s
\biggr) \biggr)
, (1.6)
where V (x) = \scrM (1 - 1/x) and \scrM (s) is GF of invariant measure of MBP that is \scrM (s) =
=
\sum
j\in \scrS
\mu js
j and
\sum
i\in \scrS
\mu iPij(t) = \mu j , j \in \scrS . Function U(y) is the inverse of V (x). The
formula (1.6) gives an alternative relation to (1.4):
q(t) =
1
U(t)
.
The following lemma is a version of more recent result that was proved in [6] (second part
statement of Lemma 1), in which the character of asymptotical decreasing of the function R(t; s)
seems to be more explicit rather than in (1.6).
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 8
1058 A. IMOMOV, A. MEYLIYEV
Lemma 1.1. If the condition (1.5) holds, then
R(t; s) =
\scrN (t)
(\nu t)1/\nu
\biggl[
1 - M(t; s)
\nu t
\biggr]
, (1.7)
where
\scrN \nu (t)\scrL
\Biggl(
(\nu t)1/\nu
\scrN (t)
\Biggr)
- \rightarrow 1 as t \rightarrow \infty . (1.8)
Here M(t; 0) = 0 for all t > 0 and M(t; s) \rightarrow \scrM (s) as t \rightarrow \infty , where \scrM (s) is GF of invariant
measures of MBP and
\scrM (s) =
1/(1 - s)\int
1
dx
x1 - \nu \scrL (x)
.
1.2. Aim and basic assumptions. The representation (1.5) implies that the second moment
2b := f \prime \prime (1 - ) = \infty . If b < \infty , then it takes place with \nu = 1 and \scrL (t) \rightarrow b as t \rightarrow \infty and we
can write asymptotic formula in type of (1.3). This circumstance suggests that we can look for some
sufficient condition such that an asymptotic relation similar to (1.3) will be true provided that (1.5)
holds. So the aim of the paper is to improve the Lemma 1.1 and thereafter to refine (1.4) and to
improve some earlier well-known results by imposing an additional condition on the function \scrL (s).
Let
\Lambda (y) := y\nu \scrL
\biggl(
1
y
\biggr)
for y \in (0, 1] and rewrite (1.5) as
[f\nu ] : f(1 - y) = y\Lambda (y).
Note that the function y\Lambda (y) is positive, tends to zero and has a monotone derivative so that
y\Lambda \prime (y)/\Lambda (y) \rightarrow \nu as y \downarrow 0 (see [3, p. 401]). Thence it is natural to write
[\Lambda \delta ] :
y\Lambda \prime (y)
\Lambda (y)
= \nu + \delta (y),
where \delta (y) is continuous and \delta (y) \rightarrow 0 as y \downarrow 0.
Throughout the paper [f\nu ] and [\Lambda \delta ] are our basic assumptions.
Since \scrL (\lambda x)/\scrL (x) \rightarrow 1 as x \rightarrow \infty for each \lambda > 0, we can write
\scrL (\lambda x)
\scrL (x)
= 1 + \varrho (x), (1.9)
where \varrho (x) \rightarrow 0 as x \rightarrow \infty . If there is some positive function g(x) so that g(x) \rightarrow 0 and
\varrho (x) = \scrO (g(x)) as x \rightarrow \infty , then \scrL (x) is said to be SV-function with remainder at infinity (see [3,
p. 185], condition SR1). As we can see below, if the function \delta (y) is known, it will be possible to
estimate a decreasing rate of the remainder \varrho (x).
Using that \Lambda (1) = \scrL (1) = a0 integration [\Lambda \delta ] yields
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 8
ON APPLICATION OF SLOWLY VARYING FUNCTIONS WITH REMAINDER . . . 1059
\Lambda (y) = a0y
\nu \mathrm{e}\mathrm{x}\mathrm{p}
y\int
1
\delta (u)
u
du.
Therefore, we have
\scrL
\biggl(
1
y
\biggr)
= a0 \mathrm{e}\mathrm{x}\mathrm{p}
y\int
1
\delta (u)
u
du.
Changing variable as u = 1/t in the integrand gives
\scrL (x) = a0 \mathrm{e}\mathrm{x}\mathrm{p}
x\int
1
\varepsilon (t)
t
dt, (1.10)
where \varepsilon (t) = - \delta (1/t) and \varepsilon (t) \rightarrow 0 as t \rightarrow \infty . It follows from (1.9) and (1.10) that
\scrL (\lambda x)
\scrL (x)
= \mathrm{e}\mathrm{x}\mathrm{p}
\lambda x\int
x
\varepsilon (t)
t
dt = 1 + \varrho (x) as x \rightarrow \infty
for each \lambda > 0, where \varrho (x) \rightarrow 0 as x \rightarrow \infty . Thus,
\lambda x\int
x
\varepsilon (t)
t
dt = \mathrm{l}\mathrm{n}[1 + \varrho (x)] = \varrho (x) +\scrO
\bigl(
\varrho 2(x)
\bigr)
as x \rightarrow \infty .
Applying the mean value theorem to the left-hand side of the last equality, we can assert that
\varepsilon (x) = \scrO (\varrho (x)) as x \rightarrow \infty . (1.11)
Thus, the assumption [\Lambda \delta ] provides that \scrL (s) to be an SV-function at infinity with the remainder
in the form of \varrho (x) = \scrO (\delta (1/x)) as x \rightarrow \infty .
1.3. Results. Our results appear due to an improvement of the Lemma 1.1 under the basic
assumptions. Let
\mho (t; s) :=
t\int
0
\delta (R(u; s))du, \mho (t) := \mho (t; 0).
Needless to say R(t; s) \rightarrow 0 as t \rightarrow \infty , due to (1.7). Therefore, since \delta (y) \rightarrow 0 as y \downarrow 0, we make
sure of
\mho (t; s)
t
=
1
t
t\int
0
\delta (R(u; s))du = o(1) as t \rightarrow \infty .
Thus, \mho (t; s) = o(t) as t \rightarrow \infty . Herewith a more important interest represents the special case when
\delta (y) = \Lambda (y). (1.12)
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 8
1060 A. IMOMOV, A. MEYLIYEV
Remark. The case (1.12) implies that \scrL (x) be an SV-function at infinity with the remainder in
the form of
\varrho (x) = \scrO
\biggl(
\scrL (x)
x\nu
\biggr)
as x \rightarrow \infty .
So, under the condition (1.12) our results appear for all SV-functions at infinity with remainder \varrho (x)
in the form above.
Theorem 1.1. Under the basic assumptions
q(t) =
\scrN (t)
(\nu t)1/\nu
\biggl(
1 - \mho (t)
\nu 2t
+ o
\biggl(
\mho (t)
t
\biggr) \biggr)
as t \rightarrow \infty , (1.13)
here and everywhere \scrN (t) is the SV-function satisfying (1.8). In addition, if (1.12) holds, then
q(t) =
\scrN (t)
(\nu t)1/\nu
\biggl(
1 - \mathrm{l}\mathrm{n} [a0\nu t+ 1]
\nu 3t
+ o
\biggl(
\mathrm{l}\mathrm{n} t
t
\biggr) \biggr)
as t \rightarrow \infty . (1.14)
Theorem 1.2. Under the basic assumptions
(\nu t)1+1/\nu P11(t) =
\scrN (t)
a0
\biggl(
1 - 1 + \nu
\nu 2
\mho (t)
t
+ o
\biggl(
\mho (t)
t
\biggr) \biggr)
(1.15)
as t \rightarrow \infty . In addition, if (1.12) holds, then
(\nu t)1+1/\nu P11(t) =
\scrN (t)
a0
\biggl(
1 - 1 + \nu
\nu 3
\mathrm{l}\mathrm{n} [a0\nu t+ 1]
t
+ o
\biggl(
\mathrm{l}\mathrm{n} t
t
\biggr) \biggr)
(1.16)
as t \rightarrow \infty .
Let \BbbP i\{ \ast \} := \BbbP \{ \ast | Z(0) = i\} and consider a conditional distribution
\BbbP \scrH (t+u)
i \{ \ast \} := \BbbP i
\bigl\{
\ast
\bigm| \bigm| t+ u < \scrH < \infty
\bigr\}
.
It was shown in [7] that the probability measure
\scrQ ij(t) := \mathrm{l}\mathrm{i}\mathrm{m}
u\rightarrow \infty
\BbbP \scrH (t+u)
i \{ Z(t) = j\} =
j
i
Pij(t) (1.17)
defines the continuous-time Markov chain \{ W (t), t \geq 0\} with states space \scrE \subset \BbbN , called the Markov
Q-process (MQP). According to the definition
\scrQ ij(t) = \BbbP i\{ Z(t) = j | \scrH = \infty \} ,
so MQP can be interpreted as MBP with non degenerating trajectory in remote future.
In a term of GF the equality (1.17) can be written as following:
Gi(t; s) :=
\sum
j\in \scrE
\scrQ ij(t)s
j = [F (t; s)]i - 1G(t; s), (1.18)
where GF G(t; s) := G1(t; s) = \BbbE
\bigl[
sW (t) | W (0) = 1
\bigr]
and
G(t; s) = - s
\partial R(t; s)
\partial s
for all t \geq 0.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 8
ON APPLICATION OF SLOWLY VARYING FUNCTIONS WITH REMAINDER . . . 1061
Combining the backward and the forward Kolmogorov equation we write it in the next form
G(t; s) = s
f(F (t; s))
f(s)
for all t \geq 0. (1.19)
Since F (t; s) \rightarrow 1 as t \rightarrow \infty uniformly for all s \in [0, 1) according to (1.18) it is suffice to
consider the case i = 1.
Theorem 1.3. Under the basic assumptions
(\nu t)1+1/\nu G(t; s) = \pi (s)\scrN (t)(1 + \rho (t; s)), (1.20)
where the function \pi (s) has an expansion in powers of s with nonnegative coefficients so that
\pi (s) =
\sum
j\in \scrE
\pi js
j and \{ \pi j , j \in \scrE \} is an invariant measure for MQP. Moreover, it has a form of
\pi (s) =
s
(1 - s)1+\nu \scrL \pi
\biggl(
1
1 - s
\biggr)
, (1.21)
where \scrL \pi (\ast ) = \scrL - 1(\ast ). Furthermore, \rho (t; s) = o(1) as t \rightarrow \infty . In addition, if (1.12) holds, then
\rho (t; s) = - 1 + \nu
\nu 3
\mathrm{l}\mathrm{n} [\Lambda (1 - s)\nu t+ 1]
t
+ o
\biggl(
\mathrm{l}\mathrm{n} t
t
\biggr)
as t \rightarrow \infty . (1.22)
Note that in accordance with Tauberian theorem for the power series (see [4, p. 513], Ch. XIII,
\S 5, Theorem 5) the relation (1.21) implies
n\sum
j=1
\pi j \sim
1
\Gamma (2 + \nu )
n1+\nu \scrL \pi (n) as n \rightarrow \infty ,
where \Gamma (\ast ) is Euler’s gamma function and (\scrL \pi \scrL )(\ast ) = 1.
Let D(t;x) := \BbbP \{ q(t)W (t) \leq x\} . In [6] (Theorem 21) it was proved, that if [f\nu ] holds, then
\mathrm{l}\mathrm{i}\mathrm{m}
t\rightarrow \infty
D(t;x) = D(x),
where
\Psi (\theta ) :=
\infty \int
0
e - \theta xdD(x) =
1
(1 + \theta \nu )1+1/\nu
.
Theorem 1.4. Let
\Delta (t; \theta ) :=
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\infty \int
0
e - \theta xdD(t;x) - \Psi (\theta )
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| .
If the basic assumptions and (1.12) hold, then
\mathrm{s}\mathrm{u}\mathrm{p}
\theta \in (0,\infty )
\Delta (t; \theta ) =
1 + \nu
\nu 3
\mathrm{l}\mathrm{n} t
t
(1 + o(1)) as t \rightarrow \infty . (1.23)
Theorem 1.4 yields that from Berry – Esseen type inequality (see [4, p. 616], Ch. XVI, \S 3,
Lemma 2) follows the following corollary.
Corollary 1.1. Under the conditions of Theorem 1.4
\mathrm{s}\mathrm{u}\mathrm{p}
x\in (0,\infty )
| D(t;x) - D(x)| = \scrO
\biggl(
\mathrm{l}\mathrm{n} t
t
\biggr)
as t \rightarrow \infty .
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 8
1062 A. IMOMOV, A. MEYLIYEV
2. Auxiliaries. The following lemma improves the statement of the Lemma 1.1.
Lemma 2.1. Under the basic assumptions
1
\Lambda (R(t; s))
- 1
\Lambda (1 - s)
= \nu t+
t\int
0
\delta (R(u; s))du. (2.1)
If, in addition, (1.12) holds, then
1
\Lambda (R(t; s))
- 1
\Lambda (1 - s)
= \nu t+
1
\nu
\mathrm{l}\mathrm{n} \nu (t; s) + o(\mathrm{l}\mathrm{n} \nu (t; s)) (2.2)
as t \rightarrow \infty , where \nu (t; s) = \Lambda (1 - s)\nu t+ 1.
Proof. From [\Lambda \delta ] we write
R\Lambda \prime (R)
\Lambda (R)
= \nu + \delta (R), (2.3)
since R = R(t; s) \rightarrow 0 as t \rightarrow \infty . By the backward Kolmogorov equation \partial F/\partial t = f(F ) and
considering representation [f\nu ] the relation (2.3) becomes
d\Lambda (R)
dt
= - \Lambda (R)
R
f(1 - R)(\nu + \delta (R)) = - \Lambda 2(R)(\nu + \delta (R)).
Therefore,
d
\biggl[
1
\Lambda (R)
- \nu t
\biggr]
= \delta (R)dt. (2.4)
Integrating (2.4) from 0 to t, we obtain (2.1).
To prove (2.2) we should calculate integral in (2.1) putting \delta (y) = \Lambda (y). Write
1
\Lambda (R(t; s))
- 1
\Lambda (1 - s)
= \nu t+
t\int
0
\Lambda (R(u; s))du. (2.5)
Since \Lambda (y) = y\nu \scrL (1/y) and R(t; s) \rightarrow 0 as t \rightarrow \infty for s \in [0, 1), the integral in the right-hand
side of (2.5) is o(t). Hence
\Lambda (R(t; s)) =
\Lambda (1 - s)
\nu (t; s)
+ o
\biggl(
\Lambda (1 - s)
\nu (t; s)
\biggr)
as t \rightarrow \infty ,
where \nu (t; s) = \Lambda (1 - s)\nu t+ 1. Therefore,
\mho (t; s) =
t\int
0
\Lambda (R(u; s))du =
1
\nu
\mathrm{l}\mathrm{n} \nu (t; s) + o(\mathrm{l}\mathrm{n} \nu (t; s)) as t \rightarrow \infty . (2.6)
This together with (2.5) implies (2.2).
Lemma 2.1 is proved.
In the proof of our results we also will essentially use the following lemma.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 8
ON APPLICATION OF SLOWLY VARYING FUNCTIONS WITH REMAINDER . . . 1063
Lemma 2.2. Let
\phi (y) := y - yK(y),
where K(y) \rightarrow 0 as y \downarrow 0. If, in addition, to the basic assumptions (1.12) holds, then
\scrL
\biggl(
1
\phi (y)
\biggr)
= \scrL
\biggl(
1
y
\biggr)
(1 +\scrO (\Lambda (y))) as y \downarrow 0. (2.7)
Proof. Since the function \scrL (x) = x\nu \Lambda (1/x) is differentiable, by virtue of the mean value
theorem we have
\scrL
\biggl(
x
1 - K
\biggr)
- \scrL (x) = \scrL \prime
\biggl(
1 - \gamma K
1 - K
x
\biggr)
K
1 - K
x, (2.8)
where K := K (1/x) and 0 < \gamma < 1. Since \varrho (x) = \scrO (\scrL (x)/x\nu ), from (1.10) and (1.11) it follows
that
\scrL \prime (u) = \scrL (u)\varepsilon (u)
u
= \scrO
\biggl(
\scrL 2(u)
u1+\nu
\biggr)
as u \rightarrow \infty . (2.9)
Denote u = (1 - \gamma K)x/(1 - K). Since K (1/x) \rightarrow 0, then u \sim x and \scrL (u) \sim \scrL (x) as x \rightarrow \infty .
Therefore after using (2.9) in the equality (2.8) and some elementary transformations the assertion
(2.7) readily follows.
Lemma 2.2 is proved.
3. Proofs of results. Proof of Theorem 1.1. Putting s = 0 in (2.1), we have
1
\Lambda (q(t))
= \nu t+
1
a0
+ \mho (t) (3.1)
and by elementary arguments we get to assertion (1.13). Similarly putting s = 0 in (2.2), we
obtain (1.14).
Theorem 1.1 is proved.
Proof of Theorem 1.2. Considering together the backward and the forward Kolmogorov equa-
tions and seeing [f\nu ], we write
\partial F (t; s)
\partial s
=
f(1 - R(t; s))
f(s)
=
R(t; s)\Lambda (R(t; s))
f(s)
.
Thence at s = 0 we deduce
P11(t) =
q(t)\Lambda (q(t))
a0
.
Hence using (1.13) and (1.14) the relations (1.15) and (1.16) easily follow.
Theorem 1.2 is proved.
Proof of Theorem 1.3. It follows from (1.19) and [f\nu ] that
G(t; s) =
R1+\nu (t; s)
f(s)
\scrL
\biggl(
1
R(t; s)
\biggr)
. (3.2)
On the other hand, (2.1) entails
R(t; s) =
\scrN (t; s)
(\nu t)1/\nu
\biggl(
1 - \mho (t; s)
\nu 2t
(1 + o(1))
\biggr)
as t \rightarrow \infty , (3.3)
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 8
1064 A. IMOMOV, A. MEYLIYEV
where \scrN (t; s) := \scrL - 1/\nu (1/R(t; s)) and \mho (t; s) =
\int t
0
\delta (R(u; s))du = o(t) as t \rightarrow \infty . From (3.3)
we conclude that
R(t; s) = q(t)
\scrN (t; s)
\scrN (t)
\biggl(
1 - \mho (t; s)
\nu 2t
(1 + o(1))
\biggr)
as t \rightarrow \infty .
Since R(t; s)/q(t) \rightarrow 1 uniformly for s \in [0, 1), then \scrN (t; s)/\scrN (t) \rightarrow 1 for all s \in [0, 1). But in
accordance with (1.9) and (1.12)
\scrL
\bigl(
R - 1(t; s)
\bigr)
\scrL (q - 1(t))
= 1 +\scrO
\biggl(
1
t
\biggr)
and, therefore,
\scrN (t; s)
\scrN (t)
= 1 +\scrO
\biggl(
1
t
\biggr)
as t \rightarrow \infty . (3.4)
Combining [f\nu ] and (3.2) – (3.4), we obtain
G(t; s) =
\pi (s)\scrN (t)
(\nu t)1+1/\nu
\biggl(
1 - 1 + \nu
\nu 2
\mho (t; s)
t
(1 + o(1))
\biggr)
as t \rightarrow \infty . (3.5)
The representation (1.20) with evanescent (1.22) follows from (2.6) and (3.5).
To show that \pi (s) is GF of invariant measure, from (1.19) we obtain the following functional
equation:
G(t+ \tau ; s) =
G(t; s)
F (t; s)
G(\tau ;F (t; s)) for all \tau > 0,
since F (t + \tau ; s) = F (\tau ;F (t; s)) (see [11]). Then taking limit as \tau \rightarrow \infty it follows from this
equation that
\pi (s) =
G(t; s)
F (t; s)
\pi (F (t; s)).
This is equivalent to the equation
\pi j =
\sum
i\in \scrE
\pi i\scrQ ij(t).
Thus, \{ \pi j , j \in \scrE \} is an invariant measure for MQP.
Theorem 1.3 is proved.
Proof of Theorem 1.4. Consider the Laplace transform
\Psi (t; \theta ) := \BbbE e - \theta q(t)W (t) = G(t; \theta (t)),
where \theta (t) = \mathrm{e}\mathrm{x}\mathrm{p}\{ - \theta q(t)\} . From [f\nu ] and (1.19) we write
\Psi (t; \theta ) = \theta (t)
\biggl(
R(t; \theta (t))
1 - \theta (t)
\biggr) 1+\nu \scrL (1/R(t; \theta (t)))
\scrL (1/(1 - \theta (t)))
. (3.6)
It follows from (2.2) that
1
\Lambda (R(t; \theta (t)))
- 1
\Lambda (1 - \theta (t))
= \nu t+
1
\nu
\mathrm{l}\mathrm{n} [\Lambda (1 - \theta (t))\nu t+ 1] + o(\mathrm{l}\mathrm{n} t) (3.7)
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 8
ON APPLICATION OF SLOWLY VARYING FUNCTIONS WITH REMAINDER . . . 1065
as t \rightarrow \infty . Since 1 - e - x \sim x - x2/2 as x \rightarrow 0, then according to our designation
\Lambda (1 - \theta (t)) = \theta \nu q\nu (t)\scrL
\biggl(
1
1 - \theta (t)
\biggr) \biggl(
1 - 1
2
\theta q(t)(1 + o(1))
\biggr) \nu
as t \rightarrow \infty . By Lemma 2.2 with K(y) = y/2
\scrL
\biggl(
1
1 - \theta (t)
\biggr)
= \scrL
\biggl(
1
q(t)
\biggr) \biggl(
1 +\scrO
\biggl(
1
t
\biggr) \biggr)
as t \rightarrow \infty . (3.8)
Then
\Lambda (1 - \theta (t)) = \theta \nu \Lambda (q(t))
\biggl(
1 +\scrO
\biggl(
1
t
\biggr) \biggr)
as t \rightarrow \infty ,
since q(t) = \scrO
\bigl(
\scrN (t)/t1/\nu
\bigr)
and \nu < 1. Thence considering (3.1)
\Lambda (1 - \theta (t)) =
\theta \nu
\nu t
\biggl(
1 - 1
\nu 2
\mathrm{l}\mathrm{n} t
t
(1 + o(1))
\biggr)
as t \rightarrow \infty . (3.9)
By using (3.9), we can write (3.7) in the following form:
1
\Lambda (R(t; \theta (t)))
= \nu t
1 + \theta \nu
\theta \nu
\biggl(
1 - 1
1 + \theta \nu
\mathrm{l}\mathrm{n} t
\nu 2t
(1 + o(1))
\biggr)
and, therefore,
R(t; \theta (t)) =
\scrN \theta (t)
(\nu t)1/\nu
\theta
(1 + \theta \nu )1/\nu
\biggl(
1 - 1
1 + \theta \nu
\mathrm{l}\mathrm{n} t
\nu 3t
(1 + o(1))
\biggr)
(3.10)
as t \rightarrow \infty , where \scrN \theta (t) := \scrL - 1/\nu (1/R(t; \theta (t))).
Since R(t; s)/q(t) \rightarrow 1 for all s \in [0, 1), then by force of (3.10) it is necessary that
R (t; \theta (t))
q(t)
- \rightarrow c(\theta ) as t \rightarrow \infty ,
where | c(\theta )| < \infty at any fixed \theta \in (0,\infty ). Therefore, according to (1.9)
\scrL
\bigl(
R - 1(t; \theta (t))
\bigr)
\scrL (q - 1(t))
= 1 +\scrO (\Lambda (q(t))) as t \rightarrow \infty (3.11)
or the same
\scrN \theta (t)
\scrN (t)
= 1 +\scrO
\biggl(
1
t
\biggr)
as t \rightarrow \infty .
Thus, (3.10) becomes
R(t; \theta (t)) =
\scrN (t)
(\nu t)1/\nu
\theta
(1 + \theta \nu )1/\nu
\biggl(
1 - 1
1 + \theta \nu
\mathrm{l}\mathrm{n} t
\nu 3t
(1 + o(1))
\biggr)
(3.12)
as t \rightarrow \infty .
Further, by using (3.8) and (3.11), we can rewrite (3.6) as
\Psi (t; \theta ) =
\biggl(
R (t; \theta (t))
1 - \theta (t)
\biggr) 1+\nu \biggl(
1 +\scrO
\biggl(
1
t
\biggr) \biggr)
as t \rightarrow \infty ,
and by using (3.12), after some transformation we obtain
\Psi (t; \theta ) = \Psi (\theta )
\biggl(
1 +
\theta \nu
1 + \theta \nu
1 + \nu
\nu 3
\mathrm{l}\mathrm{n} t
t
(1 + o(1))
\biggr)
as t \rightarrow \infty . (3.13)
The assertion (1.23) follows from (3.13).
Theorem 1.4 is proved.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 8
1066 A. IMOMOV, A. MEYLIYEV
References
1. S. Asmussen, H. Hering, Branching processes, Birkhäuser, Boston (1983).
2. K. B. Athreya, P. E. Ney, Branching processes, Springer, New York (1972).
3. N. H. Bingham, C. M. Goldie, J. L. Teugels, Regular variation, Univ. Press, Cambridge (1987).
4. W. Feller, An introduction to probability theory and its applications, vol. 2. Mir, Moscow (1967).
5. T. E. Harris, The theory of branching processes, Springer-Verlag, Berlin (1963).
6. A. A. Imomov, On conditioned limit structure of the Markov branching process without finite second moment, Malays.
J. Math. Sci., 11, № 3, 393 – 422 (2017).
7. A. A. Imomov, On Markov analogue of Q-processes with continuous time, Theory Probab. and Math. Statist., 84,
57 – 64 (2012).
8. A. N. Kolmogorov, N. A. Dmitriev, Branching stochastic process, Rep. Acad. Sci. USSR, 61, 55 – 62 (1947).
9. A. G. Pakes, Critical Markov branching process limit theorems allowing infinite variance, Adv. Appl. Probab., 42,
460 – 488 (2010).
10. E. Seneta, Regularly varying functions, Springer, Berlin (1976).
11. B. A. Sevastyanov, The theory of branching stochastic processes (in Russian), Uspekhi Math. Nauk, 6(46), 47 – 99
(1951).
12. V. M. Zolotarev, More exact statements of several theorems in the theory of branching processes, Theory Probab.
and Appl., 2, 245 – 253 (1957).
Received 31.01.19
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 8
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| id | umjimathkievua-article-684 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:03:41Z |
| publishDate | 2021 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/b1/3a5aa1dc4585e4582bcf6da63247fbb1.pdf |
| spelling | umjimathkievua-article-6842025-03-31T08:47:35Z On application of slowly varying functions with remainder in the theory of Markov branching processes with mean one and infinite variance On application of slowly varying functions with remainder in the theory of Markov Branching Processes with mean one and possibly infinite variance On application of slowly varying functions with remainder in the theory of Markov branching processes with mean one and infinite variance Imomov, A. Meyliyev , A. Imomov, Azam Imomov, A. Meyliyev , A. . UDC 519.218.2 We investigate an application of slowly varying functions (in sense of Karamata) in the theory of Markov branching processes. We treat the critical case so that the infinitesimal generating function of the process has the infinite second moment, but it regularly varies with the remainder. We improve the basic lemma of the theory of critical Markov branching processes and refine known limit results. УДК 519.218.2 Про застосування повільно змінних функцій із залишком у теорії марковських розгалужених процесів з одиничним математичним очікуванням та нескінченною дисперсією Дослiджується застосування повiльно змiнних функцiй (у сенсi Карамати) в теорiї марковських розгалужених процесiв. Критичний випадок трактується так, що iнфiнiтезимальна генеруюча функцiя процесу має нескiнченний другий момент, але регулярно змiнюється з залишком. Покращено основну лему теорiї критичних марковських розгалужених процесiв та уточнено вiдомi граничнi результати. Institute of Mathematics, NAS of Ukraine 2021-08-18 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/684 10.37863/umzh.v73i8.684 Ukrains’kyi Matematychnyi Zhurnal; Vol. 73 No. 8 (2021); 1056 - 1066 Український математичний журнал; Том 73 № 8 (2021); 1056 - 1066 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/684/9093 Copyright (c) 2021 Azam Imomov |
| spellingShingle | Imomov, A. Meyliyev , A. Imomov, Azam Imomov, A. Meyliyev , A. On application of slowly varying functions with remainder in the theory of Markov branching processes with mean one and infinite variance |
| title | On application of slowly varying functions with remainder in the theory of Markov branching processes with mean one and infinite variance |
| title_alt | On application of slowly varying functions with remainder in the theory of Markov Branching Processes with mean one and possibly infinite variance On application of slowly varying functions with remainder in the theory of Markov branching processes with mean one and infinite variance |
| title_full | On application of slowly varying functions with remainder in the theory of Markov branching processes with mean one and infinite variance |
| title_fullStr | On application of slowly varying functions with remainder in the theory of Markov branching processes with mean one and infinite variance |
| title_full_unstemmed | On application of slowly varying functions with remainder in the theory of Markov branching processes with mean one and infinite variance |
| title_short | On application of slowly varying functions with remainder in the theory of Markov branching processes with mean one and infinite variance |
| title_sort | on application of slowly varying functions with remainder in the theory of markov branching processes with mean one and infinite variance |
| topic_facet | . |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/684 |
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