On application of slowly varying functions with remainder in the theory of Markov branching processes with mean one and infinite variance

We investigate an application of slowly varying functions (in sense of Karamata) in the theory of Markov branching process. 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 process and refine known limit results.

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) = j∈S0 P 1j (t)s j it follows from (1.1) and (1.2) that the process {Z(t)} is determined by the infinitesimal GF f (s) = j∈S0 a j s j for s ∈ [0, 1). Moreover it follows from (1.2) that GF F (t; s) is unique solution of the backward Kolmogorov equation ∂F /∂t = f (F ) with the boundary condition F (0; s) = s; see [2, p. 106]. If m := j∈S ja j = f ′ (1−) is finite then F (t; 1) = 1 and due to Kolmogorov equation it can be calculated that E Z(t) |Z(0) = i = j∈S jP ij (t) = ie mt . 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] and [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 where the variable H = inf {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 ′′′ (1−) < ∞ then the following asymptotic representation holds: for all s ∈ [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 ′′ (1−) < ∞. Namely providing that = γ with index 1 < γ = 1 + α ≤ 2, he has proved that Further we assume that the infinitesimal GF f (s) has the following representation: for all s ∈ [0, 1), where 0 < ν < 1 and L(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 where V (x) = M (1 − 1/x) and M(s) is GF of invariant measure of MBP that is M(s) = j∈S µ j s j and i∈S µ i P ij (t) = µ j , j ∈ S. Function U (y) is the inverse of V (x). The formula (1.6) gives an alternative relation to (1.4): 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). Lemma 1. If the condition (1.5) holds then . 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 and thereafter to refine (1.4) and to improve some earlier well-known results by imposing an additional condition on the function L(s). Let Λ(y) := y ν L 1 y for y ∈ (0, 1] and rewrite (1.5) as Note that the function yΛ(y) is positive, tends to zero and has a monotone derivative so that yΛ ′ (y)/Λ(y) → ν as y ↓ 0; see [3, p. 401]. Thence it is natural to write where δ(y) is continuous and δ(y) → 0 as y ↓ 0.
Throughout the paper [f ν ] and [Λ δ ] are our Basic assumptions.
is said to be SV-function with remainder at infinity; see [3, p. 185, condition SR1]. As we can see below, if the function δ(y) is known it will be possible to estimate a decreasing rate of the remainder ̺(x).
Therefore we have Changing variable as u = 1/t in the integrand gives where ε(t) = −δ(1/t) and ε(t) → 0 as t → ∞. It follows from (1.9) and (1.10) that Applying the mean value theorem to the left-hand side of the last equality we can assert that Thus the assumption [Λ δ ] provides that L(s) to be an SV-function at infinity with the remainder in the form of ̺(x) = O δ 1/x as x → ∞.
1.3. Results. Our results appear due to an improvement of the Lemma 1 under the Basic assumptions. Let Needless to say R(t; s) → 0 as t → ∞, due to (1.7). Therefore, since δ(y) → 0 as y ↓ 0 we make sure of Herewith a more important interest represents the special case when δ(y) = Λ(y). (1.12) Remark. The case (1.12) implies that L (x) be an SV-function at infinity with the remainder in the form of So under the condition (1.12) our results appear for all SV-functions at infinity with remainder ̺(x) in the form above.
Let P i * := P * Z(0) = i and consider a conditional distribution It was shown in [7] that the probability measure 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: Combining the backward and the forward Kolmogorov equations we write it in the next form Since F (t; s) → 1 as t → ∞ uniformly for all s ∈ [0, 1) according to (1.18) it is suffice to consider the case i = 1. where the function π(s) has an expansion in powers of s with non-negative coefficients so that π(s) = j∈E π j s j and {π j , j ∈ E} is an invariant measure for MQP. Moreover it has a form of where L π ( * ) = L −1 ( * ). Furthermore ρ (t; s) = o(1) as t → ∞. In addition, if (1.12) holds then

Auxiliaries
The following lemma improves the statement of the Lemma 1.
In the proof of our results we also will essentially use the following lemma.
This is equivalent to the equation Thus {π j , j ∈ E} is an invariant measure for MQP.