Limiting Distributions for Multitype Branching Processes

Abstract

In this paper the asymptotic behavior of multitype Markov branching processes with discrete or continuous time is investigated in the positive regular and nonsingular case when both the initial number of ancestors and the time tend to infinity. Some limiting distributions are obtained as well as multivariate asymptotic normality is proved. The paper considers also the relative frequencies of distinct types of individuals which is motivated by applications in the field of cell biology. We obtained non-random limits for the frequencies and multivariate asymptotic normality when the initial number of ancestors is large and the time of observation increases to infinity. In fact this paper continues the investigations of Yakovlev and Yanev [32] where the time was fixed. The new obtained limiting results are of special interest for cell kinetics studies where the relative frequencies but not the absolute cell counts are accessible to measurement.

1. Introduction

This paper deals with multitype Markov branching processes with discrete or continuous time assuming a large number of ancestors. The asymptotic behavior in the positive regular and nonsingular case is investigated when both the number of ancestors N and the time t tend to infinity.

Recall that the terminology “branching processes” was first introduced by Kolmogorov and his coauthors [20, 21] considering multitype branching processes in the Markov case, which have received much attention in the literature on stochastic processes. For a further development of the theory of branching processes and their applications in biology we refer the reader to several books [1, 10, 12, 16, 18, 25, 28, 29]. Remember that the main problems of the theory of branching processes are focused on the asymptotic behavior of the probabilistic characteristics of the processes when t → ∞ assuming that the process begins from one ancestor.

Note that the multitype branching processes are considered in the present paper also from a new perspective: modeling of the relative frequencies of different types of individuals (instead of the usual counts of individuals) as functions of time. A short biological background and motivation is provided in Section 2 and the utility of relative frequencies (fractions, proportions) of distinct cell types in the analysis of biological studies of proliferation and differentiation of cells is considered. The need for such characteristics of cell kinetics arises in experimental situations where absolute cell counts are not accessible to measurement. In fact, the present article continue the investigations in this direction started in [32], where we considered the case with a large number of ancestors but the time was fixed. In this situation a limiting multivariate normal distribution was obtained (see Theorem A in Section 5) and a relevant statistical inference was developed. Now a new problem is investigated when both N and t increase to infinity and we would like to point out that the developed methods and the obtained results are quite different.

Feller [7] was probably the first who considered branching process with a large number of ancestors. For a classical Bienaymé-Galtton-Watson (BGW) process he showed a diffusion approximation in the near-critical case. Further results in this direction were obtained by Lamperti [22], Jiřina [17], Lindvall [23, 24], and Grimvall [9]. Lamperti [22] derived also some interesting limiting distributions. These results were summarized and discussed by Jagers [16].

Statistical inference for BGW processes with an increasing number of ancestors as well as limiting distributions when N and t tend to infinity were developed by Yanev [33] and Dion and Yanev [3, 4, 5] (see also a review chapter by Yanev [36]).

In the present paper some of the obtained limiting results by Lamperti [22], Yanev [33] and Dion and Yanev [3, 4, 5] are generalized for multitype Markov branching processes not only with discrete but also with continuous time. Note that usually the last two cases are investigated separately while in the present paper they are considered simultaneously. On the other hand, as usually in the theory of branching processes, the multitype generalization is quite different from the one dimensional case.

The paper is organized as follows. Section 3 introduces the basic notions and preliminary results for the further investigations. In Section 4, the probability for extinction are considered (Theorem 1) and some limit results are obtained for the relative frequencies (Theorem 2) and for the processes (Theorem 3). Section 5 deals with multivariate asymptotic normality of the processes (Theorem 4) as well as for the relative frequencies (Theorem 5). As a contrast of these results it is given Theorem A, where the obtained results in [32] are summarize. Note that in [32] we dealt with sequences of branching vectors while in the present paper we consider branching fields. In fact Theorem 2 can be interpreted as an analogue of a LLN in the case of branching arrays and Theorem 4 and Theorem 5 as analogues of a CLT in the same case. Finally in Section 6 it is shown that the obtained asymptotic results can be used to develop new statistical approach for estimation the basic parameters in branching processes with possible applications in the cell proliferation kinetics.

2. Biological Background and Motivation

The theory of branching processes has a long history of biological applications. It is worth to point out that the first asymptotic result for branching processes was obtained by Kolmogorov [19] considering some biological problems. Note that the branching processes have been proven especially useful in cell proliferation kinetics with multiple types of cells (see e.g. [29], [31], [35]).

Consider briefly the following example which is described with most details in [32]:

Oligodendrocyte type-2 astrocyte progenitor cells (referred to as O-2A progenitor cells), are known to be precursors of oligodendrocytes in the developing central nervous system. These cells grow in clones giving rise to oligodendrocytes, when they are plated in vitro and stimulated to divide by purified cortical astrocytes or by platelet-derived growth factor. Every O-2A progenitor cell can go to differentiation into an oligodendrocyte (with probability p) or it is able to proliferate (with probability 1 − p).

The first stochastic model of oligodendrocyte development in cell culture was proposed by Yakovlev et al. [30] and developed in [13, 14, 37]. The model structure was defined following a set of assumptions that specified it as a special case of the Bellman-Harris branching process with two types of cells similar to that studied by Jagers [15].

In the above-described example the scheme of observation allows to count the numbers of O-2A progenitor cells and oligodendrocytes in every of N clones but it may not be possible in many other experimental situation. As it is noted in [32], it is technically impossible to count the number of cells of a given type in the blood or bone marrow in animal experiments. Similarly in the case when studying suspension cell cultures which consist of those cell types for which no specific antibodies are available. Note that in these cases it is possible to observe only the proportions of different types of cells. This is routinely practiced in experimental and clinical laboratories but the mathematical models of cell population kinetics are traditionally formulated in terms of cell counts. That is why it is very important to investigate also the relative frequencies in multitype branching stochastic models, which is one of the motivations for the present work.

The asymptotic results obtained in this paper and in [32] give a new direction toward statistical inference and applications of branching processes in cell proliferation kinetics.

3. Multitype Markov Branching Processes with Discrete or Continuous Time

Usually branching processes with discrete or continuous time are investigated separately but in this paper we will try to treat them together. Throughout of this paper we will consider a multitype Markov branching process Z(t) = (Z₁(t), Z₂(t), …, Z_d(t)), where Z_k(t) denotes the number of cells (particles, individuals) of type k (k = 1, 2, …, d) at time t ∊ T. The time may be discrete T = N₀ = {0, 1, 2, …} or continuous T = R⁺ = [0, ∞), but the process is assumed to be positive regular and nonsingular. For most details we refer to [1, 12, 25, 28].

Further on we will use the following notions:

$$\begin{array}{l}{\delta }_i=({\delta }_{i1},\dots ,{\delta }_{ij},\dots ,{\delta }_{id}),{\delta }_{ij}=1\mathrm{for}i=j,{\delta }_{ij}=0\mathrm{for}i\ne j,i,j=1,\dots ,d\text{;}\\ \mathbf{\text{S}}=(s_1,\mathrm{\dots },s_d),|\mathrm{s}|\le 1=(1\text{,}\dots ,1),\mathrm{\alpha }=({\mathrm{\alpha }}_1,\mathrm{\dots },{\mathrm{\alpha }}_d),{\mathbf{\text{S}}}^{\mathrm{\alpha }}={\displaystyle \prod _{k=1}^d{S}_k^{{\alpha }_k}.}\end{array}$$

Introduce the probability generating functions (p.g.f.)

F_i(t; s) = IE{s^Z(t) | Z(0) = δ_i}, i = 1, 2, …, d; F(t; s) = (F₁(t; s), …, F_d(t; s)).

Remember that in the Markov case

$$\mathbf{\text{F}}(t+\tau ;\mathbf{s})=\mathbf{\text{F}}(t;\mathbf{\text{F}}(\tau \text{;}\mathbf{s}\text{)), }t,\tau \in \mathbf{T}\text{,}$$ (1)

which has the following stochastic analog

$$\mathbf{Z}\text{(t + }\tau \text{) = }{\displaystyle \sum _{i=1}^d{\displaystyle \sum _{j=1}^{{\text{Z}}_i(t)}{\mathbf{\text{Z}}}^{(i)}}}(\tau ;j),$$

where the vectors {Z⁽ⁱ⁾(τ; j)}_j≥1 are i.i.d. as Z⁽ⁱ⁾(τ) and Z⁽ⁱ⁾(τ) means Z(τ) with Z(0) =δ_i.

In the discrete-time case we will use the offspring p.g.f.

$${\text{h}}_i(\mathbf{\text{s}})=F_i(1;\mathbf{\text{s}})={\displaystyle \sum _{\mathrm{\alpha }\in {\mathbf{\text{R}}}^d}{p}_i(\mathrm{\alpha }){\mathbf{\text{s}}}^{\mathrm{\alpha }}},h_i(1)=1,i=1,2,\dots ,d;\mathbf{h}\text{(}\mathbf{s}\text{) =(}{h}_1(\mathbf{s}\text{),}\dots \text{,}{h}_d(\mathbf{s})).$$

Then by (1) one has

$$\mathbf{\text{F}}(t+1;\mathbf{\text{s}})=\mathbf{\text{F}}(t;\mathbf{\text{h}}(\mathbf{\text{s}}))=\mathbf{h}(\mathbf{\text{F}}(t;\mathbf{\text{s}})),t=0,1,2,\dots ,$$ (2)

which means that F(t; s) is the t-th iterate of h(s).

In the continuous-time case as t ↓ 0

F(t; s) =s + tf(s) + o(t) uniformly for |s| ≤ 1,

where f(s) = (f₁(s), …,f_d(s)) is the vector of the infinitesimal generating functions

$$f_i(\mathbf{\text{s}})={\displaystyle \sum _{\mathrm{\alpha }\in {\mathbf{\text{R}}}^d}{r}_i(\mathrm{\alpha }){\mathbf{\text{s}}}^{\mathrm{\alpha }}},f_i(\mathbf{1})=0,i=1,2,\dots ,d.$$

Then instead (2) the Kolmogorov backward and forward equations hold

$$\frac{\partial \mathbf{\text{F}}(t;\mathbf{\text{s}})}{\partial t}=\mathbf{\text{f}}(\mathbf{\text{F}}(t;\mathbf{\text{s}})),\frac{\partial \mathbf{\text{F}}(t;\mathbf{\text{s}})}{\partial t}={\displaystyle \sum _{i=1}^d{f}_i(\mathbf{\text{s}})}\frac{\partial \mathbf{\text{F}}(t;\mathbf{s})}{\partial s_i},\mathbf{F}\text{(0; }\mathbf{s}\text{)}=\mathbf{s}\text{.}$$

It is well-known that all characteristics of the process can be obtained from the p.g.f. In particular, the following notation will be used:

$$A_{ij}(t)=\text{IE{}{Z}_j(t)|\mathbf{Z}\text{(0)}={\delta }_i\}=\frac{\partial }{\partial s_j}{F}_i(t;\mathbf{\text{s}}){|}_{\text{s=}\mathbf{1}},$$

$$A_{ij}=\frac{\partial h_i(\mathbf{\text{s}})}{\partial s_j}{|}_{\mathbf{\text{s}}=1},a_{ij}=\frac{\partial f_i(\mathbf{\text{s}})}{\partial s_j}{|}_{\mathbf{s}\text{=}1},i,j=1,2,\dots ,d.$$

Introduce the matrices A(t) = ‖A_ij(t)‖, A = ‖A_ij|| and a =‖a_ij‖. Then

$$\mathbf{\text{A}}(t)={\mathbf{\text{A}}}^t,t\in {\mathbf{\text{N}}}_0;\mathbf{\text{A}}(t\text{)}={\mathbf{\text{e}}}^{\text{a}t}={\displaystyle \sum _{n=0}^{\infty }{\mathbf{\text{a}}}^n{t}^n/n!,t\in {\mathbf{\text{R}}}^{+}.}$$

Similarly one can obtain

$$B_{jk}^{(\mathrm{i}\text{)}}(t)=\text{IE}\{{\text{Z}}_j(t)[Z_k(t)-{\delta }_{jk}]|\mathbf{Z}\text{(0) = }{\delta }_{\text{i}}\}=\frac{{\partial }^2}{\partial s_j\partial s_k}{F}_i(t;\mathbf{\text{s}})\text{|}{}_{\mathbf{\text{s}}=1}\text{,}$$

$$\begin{array}{ccc}{B}_{jk}^{(i)}=\frac{{\partial }^2{h}_i(\mathbf{\text{s}})}{\partial s_j\partial s_k}{|}_{\mathbf{\text{s}}=1}\text{, }& b_{jk}^{(\mathrm{i})}=\frac{{\partial }^{2}f\text{i}(\mathbf{\text{s}})}{\partial {\text{s}}_j\partial s_k}{|}_{\mathbf{\text{s}}=1}\text{;}& i,j,k\text{=1, 2, }\dots \text{,}d\text{.}\end{array}$$

Further on we will consider the positive regular and nonsingular case and then we will denote the Perron-Frobenius eigenvalue by R for the matrix A and by r for the matrix a. In both cases we will use one and same notion for the corresponding right u =(u₁, …, u_d) and left v = (υ₁, …, υ_d) eigenvectors. It is well-known that they can be chosen positive and normalized such that ${\displaystyle \sum _{i=1}^d}{u}_i=1,{\displaystyle \sum _{i=1}^d{u}_i{\upsilon }_i=1.}$ Then it is well-known that as t → ∞

$$\begin{array}{cc}{A}_{ij}(t)\sim u_i{\upsilon }_j{R}^t,t\in {\mathbf{\text{N}}}_0;& A_{ij}(t)\sim u_i{\upsilon }_j{\mathbf{\text{e}}}^{rt},t\in {\mathbf{\text{R}}}^{+};\end{array}$$ (3)

Note that the moment structure of multitype Galton-Watson process is investigated in Quine [27] where some linear recurrence relations are derived.

We will need also the following notation

$$B(t)={\displaystyle \sum _{i,j,k=1}^d{\upsilon }_i{B}_{jk}^{(i)}(t)u_j{u}_k,B=}{\displaystyle \sum _{i,j,k=1}^d{\upsilon }_i{B}_{jk}^{(i)}{u}_j{u}_k,b={\displaystyle \sum _{i,j,k=1}^d{\upsilon }_i{b}_{jk}^{(i)}{u}_j{u}_k.}}$$ (4)

The main purpose of this paper is the investigation of the asymptotic behavior of the processes with a large number of ancestors. For simplicity (and without any essential restriction) we will assume that Z(0) = (N, 0, …, 0) and Z₁(0) = N → ∞. In this case we will use the notation Z(t; N) = (Z₁(t; N), Z₂(t; N), …, Z_d(t; N)), t ∊ T, and we will consider the situation when both N and t go to infinity simultaneously. In general, N and t are free parameters, but further on one can assume that N = N(t) and N(t) ↑ ∞ as t → ∞.

By the branching property it follows that

$$\begin{array}{cc}{Z}_i(t;N)={\displaystyle \sum _{k=1}^N{Z}_{i,k}(t),}& i=1,2,\dots ,d,\end{array}$$ (5)

where ${\{Z_{i,k}(t)\}}_{k=1}^N$ are i.i.d. copies of the branching process Z_i(t) with Z(0) = δ₁.

In fact, (5) is equivalent to the following relation

$$F_N(t;\mathbf{\text{s}})={\text{IE{}\mathbf{s}}^{\mathbf{\text{z }}(t\text{)}}|\mathbf{Z}\text{(0)}=(N\text{, 0, }\dots \text{,}\text{0)}}=F^N(t;\mathbf{s}\text{),}$$ (6)

where $F(t;s){\text{= }F}_1(t\text{; }\mathbf{s})={\text{IE{}\mathbf{s}}^{\mathbf{\text{z }}(t\text{)}}\text{|}\mathbf{Z}\text{(0) =}{\delta }_1\}.$

Introduce the total number of cells at the moment t as

$$U(t)={\displaystyle \sum _{k=1}^d{Z}_k(t),U}(0)=Z_1(0)=1,t\in \mathbf{T}\text{.}$$ (7)

Then by (5) and (7) it follows that

$$U(t\text{;}N)={\displaystyle \sum _{i=1}^d{Z}_i(t;N)={\displaystyle \sum _{k=1}^N{U}^{(k)}}(t),}$$ (8)

where $U^{(k)}(t)={\displaystyle \sum _{i=1}^d{Z}_{i,k}(t)},k\ge 1,$ are i.i.d. as U(t).

The relative frequencies (fractions, proportions) of types can be defined on the non-extinction set {U (t; N) > 0} as follows:

$$\begin{array}{cc}{\Delta }_i(t;N)=Z_i(t;N)/U(t;N),& i=1,2,\dots ,d,\end{array}$$ (9)

with the obvious condition

$${\displaystyle \sum _{k=1}^d{\Delta }_k(t;N)=1.}$$ (10)

The investigation of the relative frequencies is very important for the applications (especially in the cell biology) because there are a lot of situations when it is not possible the observe the numbers of individuals but only their relative proportions.

In what follows, we will also need the following deterministic proportions:

$$\begin{array}{cc}{p}_i(t)=A_{1i}(t)/M(t),& i=1,2,\dots ,d;t\in \mathbf{T}\text{,}\end{array}$$ (11)

where

$$M(t)=\text{IE{}U(t\text{)}}={\displaystyle \sum _{j=1}^d{A}_{1j}(t).}$$ (12)

4. Probability of Extinction and Limiting Distributions

In this section we will consider critical processes with R = 1, 0 < B < ∞, or r = 0, 0 < b < ∞, and non-critical processes (R ≠ 1 or r ≠ 0) with the classical” X log X condition”.

Theorem 1

Let q(t; N) = Pr{U(t; N) = 0} and N, t → ∞.

1. If {R > 1 or r > 0} ∨ {R = 1 or r = 0 and N/t → ∞} ∨ {R < 1, N R^t → ∞ or r < 0, Ne^rt → ∞} then q(t; N) → 0 (non-extinction);

2. If {R = 1 or r = 0 and N/t → 0} ∨ {R < 1, NR^t → 0 or r < 0, Ne^rt → 0 then q(t; N) → 1 (extinction);

3. If 0 < C < ∞ and {R = 1 or r = 0 and N/t → C} ∨ {R < 1, N R^t → C or r < 0, Ne^rt → C} then q(t; N) → q, 0 < q < 1 (extinction with probability q or non-extinction with probability 1 − q).

Remark 1

The constants in the critical and subcritical cases as well as for the discrete or continuous processes can be different but in (iii) and further on we will use for simplicity one and the same letter (without any indices).

Proof

Note that by (6) one has q(t; N) = F_N(t; 0) = q^N(t), where

q(t) = Pr{U(t) = 0} = F(t; 0), 0 =(0, 0, …, 0),

is the extinction probability and q(t) < 1 for every fixed t.

Therefore logq(t; N) = N log[1 − Q(t)], where Q(t) = 1 − q(t).

1) Let R < 1 or r < 0. Then Q(t) ~ Ku₁R^t or Q(t) ~ Ku₁e^rt as t → ∞ where K are some positive constants.

Therefore

$$\text{log }q(t;N)~-K{u}_{\text{1}}N{R}^t\text{ or log }q(t;N)~-K{u}_{\text{1}}N{\mathbf{\text{e}}}^{\mathit{rt}}.$$

Hence

$$\begin{array}{c}q(t;N)\to 0\text{ if }N{R}^t\to \infty (N{\mathbf{\text{e}}}^{rt}\to \infty ),\hfill \\ q(t;N)\to \text{1 if }N{R}^t\to 0(N{\mathbf{\text{e}}}^{rt}\to 0)\text{ and}\hfill \\ q(t;N)\to q\text{ if }N{R}^t\to C(N{\mathbf{\text{e}}}^{rt}\to C),0<C<\infty ,\text{ where }0<q={\text{e}}^{-K{u}_{\text{1}}C}<\text{ 1 }.\hfill \end{array}$$

2) Let R = 1 or r = 0. In this case Q(t) ~2u₁/Bt or Q(t) ~ 2u₁/bt as t → ∞.

Then

$$\text{log }q(t;N)~-\text{2}{u}_{\text{1}}N/Bt\text{ or log }q(t;N)~-\text{2}{u}_{\text{1}}N/bt.$$

Therefore one obtains

$$\begin{array}{ccc}q(t;N)\hfill & \hfill \to \hfill & 0\text{ if }N/t\to \infty ,\hfill \\ q(t;N)\hfill & \hfill \to \hfill & \text{1 if }N/t\to 0\text{ and}\hfill \\ q(t;N)\hfill & \hfill \to \hfill & q\text{ if }N/t\to C,0<C<\infty ,\text{ where }q={\text{ e}}^{-\text{2}{u}_{\text{1}}C/B}\text{ or }q={\mathbf{\text{ e}}}^{-\text{2}{u}_{\text{1}}C/b}\text{ and }0<q<1.\hfill \end{array}$$

3) Let R > 1 or r > 0. In this case q(t) ↑ q as t → ∞. Therefore q(t; N) = q^N(t) → 0 as N → ∞ uniformly for 0 < t ≤ ∞.

Theorem 2

Let p_k = υ_k/V, k = 1, …, d, $V={\displaystyle \sum _{k=1}^d{\upsilon }_k},$ and N, t → ∞.

1. If {R < 1, NR^t → ∞ or r < 0, Ne^rt → ∞} ∨ {R = 1 or r = 0 and N/t → ∞} then Δ_k(t; N) → p_k in probability, k = 1, 2, …, d;

2. If {R > 1 or r > 0} then Δ_k(t; N) → p_k a.s., k = 1, 2, …, d.

Proof

Remember that by (6)

$${\text{IE{}\mathbf{e}}^{-\mathrm{\lambda }{Z}_k(t;N)}\}=F_{(k)}^N \; (t;{\mathbf{\text{e}}}^{-\mathrm{\lambda }}),k=1 ,2 ,\mathrm{\dots } ,d.$$ (13)

where F _(k) (t; s_k) = F (t ; 1, …, 1, s_k, 1, …, 1).

1) Let first consider the subcritical case R < 1 with NR^t → ∞ (t $\in$N₀) or r < 0 with Ne^rt → ∞ (t $\in$R⁺).

Denote by β_(k) (t, N ; λ), λ > 0, anyone of the following Laplace transforms:

$$\text{IE}\left\{{\mathbf{\text{e}}}^{-\mathrm{\lambda }{Z}_k(t;N)/N{R}^t}\right\}=F_{(k)}^N(t;{\mathbf{\text{e}}}^{-\mathrm{\lambda }/N{R}^t}) ,t\in {\mathbf{\text{N}}}_0 ,\text{or}$$

$$\text{IE }\left\{{\mathbf{\text{e}}}^{-\mathrm{\lambda }{Z}_k(t;N)/N{\text{e}}^{rt}}\right\}=F_{(k)}^N(t;{\mathbf{\text{e}}}^{-\mathrm{\lambda }/N{e}^{rt}}) ,t\in {\mathbf{\text{R}}}^{+}\text{.}$$

Consider first the discrete-time case for which it is known that (see Athreya and Ney [1972, Ch. V. 3], Mode [1971,Ch. 1.11] or Sevastyanov [1971, Ch. VI. 2])

$$\text{1 }-F_{(K)}(t;0)\backsim K{u}_{\text{1}}{R}^t,0<K<\infty ,$$

$$\{\text{ lim}\{[1-F_{(k)}(t;\mathbf{\text{s}})]/[1-F_{(k) }(t;0)]\}=1- F_{(k)}^{\ast }(\mathbf{\text{s}}) ,A_k^{\ast }=\frac{\partial }{\partial s_k}{F}_{(k)}^{\ast }(\mathbf{\text{s}}) {|}_{\mathbf{s}\text{=1}}={\upsilon }_k/K.$$ (14)

Since

log β_(k)(t, N; λ) = N logF_(k)(t; e^−λ/NRt) ~ −N{1 − F_(k)(t; e^−λ/NRt)}

then applying (14) one obtains

$$\text{log }{\beta }_{(k)}(t,N;\mathrm{\lambda })~-K{u}_{1}N{R}^t\{1-F_{(k)}^{\ast } ({\mathbf{e}}^{-\lambda /{\mathit{NR}}^t})\}.$$ (15)

On the other hand

$$\text{lim}\{[1-F_{(k) }^{\ast }(t;{\mathbf{\text{e}}}^{-\mathrm{\lambda }/N{R}^t})]/[1-{\mathbf{\text{e}}}^{-\mathrm{\lambda }/N{R}^t}]\}=A_k^{\ast }\text{ and}1-{\mathbf{\text{e}}}^{-\mathrm{\lambda }/N{R}^t}~\mathrm{\lambda }/N{R}^t.$$

Therefore by (15) one has log β_(k)(t, N; λ) ~ − λu₁υ_k and by the continuity theorem for Laplace transforms as NR^t → ∞ one obtains

$$Z_k(t;N)/N{R}^t\to u_1{\upsilon }_k\mathit{in}\mathit{probability},k=1,\dots ,d.$$ (16)

Then from (8) and (16) it follows

$$U(t;N)/N{R}^t\to u_{1}V\mathit{ in }\mathit{probability},V={\displaystyle \sum _{k=1}^d{\upsilon }_k .}$$ (17)

Finally from (9), (16) and (17) it is not difficult to see that in the case R < 1, NR^t → ∞

$${\Delta }_k(t;N)\to p_k={\upsilon }_k/V\mathit{ in }\mathit{probability},k=1,2,\dots ,d.$$ (18)

In the continuous-time case the following relations hold (similarly to (14))

$$1-F_{(k)}(t;0)\sim K{u}_1{\mathbf{\text{e}}}^{rt},0<K<\infty ,$$

$$\{\lim \{[1-F_{(k)} (t;\mathbf{\text{s}})]/[1-F_{(k)} (t;0)]\}=1-F_{(k)}^{\ast }(\mathbf{\text{s}}),{\displaystyle \sum _{i=1}^d{f}_i(\mathbf{\text{s}})\frac{\partial }{\partial s_i}}{F}_{(k)}^{\ast }(\mathbf{\text{s}})=-r[1-F_{(k)}^{\ast }(\mathbf{\text{s}})] .$$ (19)

By differentiating the last relation in (19) and putting s = 1one obtains

$$r{A}_k^{\ast }={\displaystyle \sum _{i=1}^d{A}_i^{\ast }{a}_{ik}},k=1,\dots ,d.$$

Therefore by the Frobenius Theorem it follows that $A_k^{\ast }=C_1{\upsilon }_k$ for some constant C₁. Now using a similar arguments as in the proof of the discrete case (see Theorem 6.2.4 in Sevastyanov, 1971) it is not difficult to obtain that in fact C₁ = 1/ K. On the other hand, by (3) and the first relation from (19) onehas $\text{IE{}{Z}_k(t)|Z_k(t)>0,\mathbf{Z}\text{(0)}={\delta }_1\}=A_{1k}(t)/[1-F_{(k)}(t;0)]\to {\upsilon }_k/K=A_k^{\ast } .$

Hence applying (19) one obtains

$\log {\beta }_{(k)}(t,N;\mathrm{\lambda })=N\log F_{(k)}^N(t;{\mathbf{\text{e}}}^{-\mathrm{\lambda }/N{e}^{rt}})\sim -N\{1-F_{(k)}^N(t;{\mathbf{\text{e}}}^{-\mathrm{\lambda }/N{e}^{rt}})\}$

and similarly to the discrete-time case

$$\mathrm{log }{\beta }_{(k)}(t,N;\mathrm{\lambda })\sim -K{u}_{1}N{e}^{rt}\{1-F_{(k)}^{\ast }({\mathbf{\text{e}}}^{-\mathrm{\lambda }/N{e}^{rt}})\},$$ (20)

where $\lim \{[1-F_{(k)}^{\ast }(t;{\mathbf{\text{e}}}^{-\mathrm{\lambda }/N{e}^{rt}})]/[1-{\mathbf{\text{e}}}^{-\mathrm{\lambda }/N{e}^{rt}}]\}=A_k^{\ast }={\upsilon }_k/K\text{and}\text{1}-{\mathbf{\text{e}}}^{-\mathrm{\lambda }/N{e}^{rt}}\sim \mathrm{\lambda }/N{\mathbf{\text{e}}}^{rt}.$

Finally from (20) one obtains that log β_(k)(t, N; λ) ~ − λu₁υ_k and by the continuity theorem as Ne^rt → ∞ one has

$$Z_k(t;N)/N{\mathbf{\text{e}}}^{rt}\to u_1{\upsilon }_{k}in\mathit{probability},k=1,\dots ,d.$$ (21)

Hence by (8) and (21) it follows

$$U(t;N)/N{\mathbf{\text{e}}}^{rt}\to u_{1}Vin\mathit{probability},V={\displaystyle \sum _{k=1}^d{\upsilon }_k .}$$ (22)

Then from (9), (21) and (22) it is not difficult to see that in the case r < 0, Ne^rt → ∞

$${\Delta }_k(t;N)\to p_k={\upsilon }_k/Vin\mathit{probability},k=1,2,\dots ,d.$$ (23)

2) Consider now the critical case (R = 1 or r = 0) with N/t → ∞.

Using (13) one can introduce the following Laplace transform

$${\gamma }_{\text{(}k\text{)}} (t,N;\mathrm{\lambda })={\text{IE{}\mathbf{e}}^{-\mathrm{\lambda }{Z}_k(t;N)/N}\}=F_{(k)}^N(t;{\mathbf{\text{e}}}^{-\mathrm{\lambda }/N}) ,t\in \mathbf{T}\text{.}$$ (24)

Then $\log {\gamma }_{(k)}(t,N;\mathrm{\lambda })=N\log F_{(k)}(t;{\mathbf{\text{e}}}^{-\mathrm{\lambda }/N})\sim N\{1-F_{(k)}(t;{\mathbf{\text{e}}}^{-\mathrm{\lambda }/N})\}.$

As t → ∞ in the discrete-time case one has (see Athreya and Ney [1972, Ch. V. 5], Mode [1971,Ch. 1.10] and Sevastyanov [1971, Ch. VI. 3])

$$1-F(t;\mathbf{\text{s}})\sim \{u_1{\displaystyle \sum _{j=1}^d{\upsilon }_j(1-s_j)\}/\{1+\frac{B}{2}t{\displaystyle \sum _{l=1}^d{\upsilon }_l(1-s_l})\}}$$ (25)

uniformly for 0 ≤ s ≤ 1, s ≠ 1, where B is defined in (4).

Therefore by (13) and (23) one obtains

$$\log {\gamma }_{(k)}(t,N;\mathrm{\lambda })\sim -N{u}_1{\upsilon }_k(1-{\mathbf{\text{e}}}^{-\mathrm{\lambda }/N})/\{1+\frac{B}{2}t{\upsilon }_k(1-{\mathbf{\text{e}}}^{-\mathrm{\lambda }/N})\}\sim -u_1{\upsilon }_k\mathrm{\lambda }/(1+\frac{B}{2}{\upsilon }_k\mathrm{\lambda }\frac{t}{N}) .$$ (26)

Since γ_(k)(t, N; λ) → e^−λu1υk as N/t → ∞ then by the continuity theorem it follows that

$$Z_k(t;N)/N\to u_1{v}_{k}in\mathit{probability},k=1,\dots ,d.$$ (27)

Hence by (8) and (27) one obtains

$$U(t;N)/N \to u_{1}Vin\mathit{probability},V={\displaystyle \sum _{k=1}^d{\upsilon }_k} .$$ (28)

Finally by (9), (27) and (28) it follows that in the case R < 1, NR^t → ∞

$${\Delta }_k(t;N)\to p_k={\upsilon }_k/V\mathit{ in }\mathit{probabality},k=1,2,\dots ,d.$$ (29)

Note that some results obtained for discrete-time Markov branching processes can betransferred to the continuous-time case using the method of embedded branching processes. In fact,for F(t; s), t ∊R⁺, and ε_n = 2⁻ⁿ, n = 1, 2, …, one canconsider the p.g.f. of the embedded process $\stackrel{\sim }{F}(t;\mathbf{\text{s}})=F(k{\epsilon }_n;\text{s})\text{ =}{F}_n(k;\mathbf{\text{s}}),$ where t ∊ { k∊_n, k = 0, 1, 2, …}. In thiscase $R_n={\mathbf{\text{e}}}^{r{\epsilon }_n}$ and for the critical processes B(ε_n) = bε_n (1+ o(1)) as n → ∞. Therefore the asymptotic relations (25) and (26) will be valid also in the continuous-time case where B can be replaced by b from (4). Hence (27) – (29) hold.

3) In the supercritical case (R > 1 or r > 0) it is well-known (see Athreya and Ney [1972, Ch. V. 6 & 7], Mode [1971,Ch. 1.8 and 1.9] or Sevastyanov [1971, Ch. VI. 4]) that for W_k(t) = Z_k(t)/R^t, t ∊ N₀, or W_k(t) = Z_k(t)/e^rt, t ∊ R⁺, the following relations hold as t → ∞

$$W_k(t) \to {\upsilon }_{k}Wa.s.,k=1,\dots ,d;\text{ IE}\{W| \mathbf{\text{Z}}(0)={\delta }_1\}=u_1.$$ (30)

Let η_k (t; N)= Z_k (t; N)/NR^t, t ∊ N₀, or η_k (t; N)= Z_k (t; N)/Ne^rt, t ∊ R⁺. Then by (5) one has ${\eta }_k(t;N)=\frac{1}{N}{\displaystyle \sum _{j=1}^N{W}_{k,j}(t),}$ where ${\{W_{k,j}(t)\}}_{j=1}^N$ are i.i.d. as W_k(t), k = 1, …, d.

On the other hand, as N, t → ∞

$${\eta }_k(t;N)={\upsilon }_{k}W+\frac{1}{N}{\displaystyle \sum _{j=1}^N{\zeta }_{k,j}(t)\to {\upsilon }_{k}Wa.s.,k=1,\dots ,d,}$$

where ζ _k,j(t) =W_k,j (t)−υ_kW and by (30) one has that $\frac{1}{N}{\displaystyle \sum _{j=1}^N{\zeta }_{k,j}(t)\to }0\mathit{\text{ a.s.}}$

Therefore by (8) one obtains that U(t; N)/NR^t, t ∊ N₀, or U(t; N)/Ne^rt, t ∊ R⁺, converges a.s. to VW as N, t → ∞, $V={\displaystyle \sum _{k=1}^d{\upsilon }_k}.$ Then the statement (ii) follows.

Remark 2

In other words, the fractions Δ_i(t; N), i = 1, …, d, are consistent estimators for p_i under the conditions of Theorem 2. By virtue of the fact that 0 ≤ Δ_i (t; N) ≤ 1 and the dominated convergence theorem (DCT), it follows that IE{Δ_i (t; N) } converges to p_i, implying that Δ_i (t; N) is an asymptotically unbiased estimator for p_i, i = 1, 2, …, d. Similarly, one has that $\text{IE}\{{\Delta }_i^2(t;N)\}\to p_i^2$ by the DCT and therefore Var {Δ_i (t; N) } → 0

Note also that for each i the defined in (11) quantity p_i (t) may be interpreted as a probability for a randomly chosen cell attime t to be of the type i and by (3), (11) and (12) it is not difficult to see that p_i (t) → p_i as $t\to \infty ,{\displaystyle \sum _{i=1}^d{p}_i=1.}$

On the other hand, if t ∊ T is fixed and N → ∞ then from (5) and (8) applying the law of large numbers (LLN) one obtains

$${△}_i(t;N)=[\frac{1}{N}{\displaystyle \sum _{k=1}^N{Z}_{i,k}(t)]/[\frac{1}{N}{\displaystyle \sum _{k=1}^N{U}^{(k)}(t)]\to p_i(t)}}a.s.,i=1,\dots ,d.$$

Put it another way, the frequencies Δ_i(t; N), i = 1, …, d, are strongly consistent estimators for p_i(t) when considered as functions of the initial number of ancestors N (but the time t is fixed). Moreover, in this case the frequencies are also asymptotically normal (see Theorem A in the next section and Remark 3).

Theorem 3

Let N, t → ∞.

(i)If {R < 1, NR^t → 0 or r < 0, Ne^rt → 0} ∨ {R = 1 or r = 0 and N/t → 0} then Z_k(t; N) → 0 in probability, k = 1, 2, …, d.

(ii) If { R < 1, NR^t → C or r < 0, Ne^rt → C and 0 < C < ∞ } then $Z_k(t;N)\stackrel{d}{\to }{\chi }_k$, k = 1, 2, …, d, and $U(t;N)\stackrel{d}{\to }\chi \text{,}$ where

$${\phi }_k(\mathrm{\lambda }) =\text{ IE}\{e^{-\mathrm{\lambda }\chi \kappa }\}=\text{ exp}\{-u_{1}CK[1-F_{(k)}^{\ast }({\mathbf{\text{e}}}^{-\mathrm{\lambda }/C})]\},$$

$$\phi (\mathrm{\lambda }) =\text{ IE}\{e^{-\mathrm{\lambda }\chi }\}=\text{ exp}\{-u_{1}CK[1-F^{\ast }({\mathbf{\text{e}}}^{-\mathrm{\lambda }/C})]\}.$$

The p.g.f. $F_{(k)}^{\ast }(s)$and F*(s)are well defined in the discrete-time case by (14) or by (19) in the continuous-time case.

(iii)If {R = 1 or r = 0 and N/t → C, 0 < C < ∞ } then Z_k(t; N)/N $\stackrel{d}{\to }$ϱ_k, k = 1, 2, …, d, and U(t; N)/N $\stackrel{d}{\to }$ϱ, where

$${\psi }_k(\mathrm{\lambda })=\text{ IE}\{e^{-\mathrm{\lambda }\varrho k}\}= \text{exp }\{-\mathrm{\alpha }[1-{\beta }_k/({\beta }_k+\mathrm{\lambda })]\},$$

$$\psi (\mathrm{\lambda }) =\text{ IE}\{e^{-\mathrm{\lambda }\varrho }\} =\text{exp }\{-\mathrm{\alpha }[1-\beta /(\beta +\mathrm{\lambda })]\}$$

$$\mathit{and} \mathrm{\alpha }= 2u_1/C,{\beta }_k= 2/CB{\upsilon }_{k}or{\beta }_{k }= 2/Cb{\upsilon }_k,{\beta }^{-1}={\displaystyle \sum }_{k=1}^d{\beta }_k^{-1}.$$

Proof

(i) In this case for the subcritical processes by (15) and (20) it is not difficult to obtain that log β_(k)(t, N; λ) →0 and for the critical processes log γ_(k)(t, N; λ) → 0 by (26), which means that IE{e^−λZk(t;N)} → 1, showing the statement by the continuity theorem.

(ii) This case follows immediately from (15) and (20). Note that limiting distributions are Compound Poisson, i.e. ${\chi }_k={\displaystyle \sum _{j=1}^n{\xi }_k(j)},$ where η has a Poisson distribution with parameterα = u₁CK and ξ _k (j), j ≥ 1, are i.i.d. with a Laplace transform $\text{IE{}{e}^{-\mathrm{\lambda }{\xi }_k(j)}\text{}=}{F}_{(k)}^{\ast }(e^{-\mathrm{\lambda }/C}) .$ Similarly for χ.

(iii) The statement follows directly from (26) (with a remark that in the continuous-time case one can use b instead B). The limiting distributions are also Compound Poisson but now ${\varrho }_k={\displaystyle \sum _{j=1}^{\eta }{\xi }_k(j)}$ with η ∊ Po(α), α = 2u₁/C, and {ξ_k(j)} and i.i.d. and exponentially distributed with parameter β_k = 2/CBυ_k or β_k = 2/Cbυ_k. Similarly for $\varrho ={\displaystyle \sum _{j=1}^n{\xi }_k,}$ where {ξ_k} are i.i.d. and ξ_k ∊ Exp(β), β = 2/BKV or β =2/bKV, $V={\displaystyle \sum _{k=1}^n\upsilon k}.$

5. Multivariate Asymptotic Normality

In this section we will assume that the processes have finite second moments.

Consider X(t; N) = (X₁(t; N), …, X_d(t; N)), t ∊ T, where for some norming functions D_k(t)

$$X_k(t;N)=[Z_k(t;N)-A_{1k}(t;N)]/[D_k(t)\sqrt{N}]=\frac{1}{\sqrt{N}}{\displaystyle \sum _{j=1}^N [Z_{k,j}(t)-A_{1k}(t)]/D_k(t)]} ,$$ (31)

because of (5) and the fact that A_1k(t; N) = IE{ Z_k(t; N) } = N A_1k (t) , k = 1, …, d.

Theorem 4

Assume N, t → ∞.

(i) If {R < 1, NR^t→ ∞, or r < 0,Ne^rt → ∞}and $D_k(t)\equiv D(t)\sim \{\sqrt{R^t}\mathit{ or }\sqrt{{\text{e}}^{rt}}\}$ for k = 1, …, d,then

$$\mathbf{\text{X}} (t;N)\stackrel{d}{\to }\xi =(\xi ,\dots ,{\xi }_d),$$ (32)

and ξ = (ξ₁, …, ξ_d) has a multivariate normal distribution with IEζ_i = 0 and a covariance matrix C =‖C_jk‖, where $C_{jk}=\lim B_{jk}^{(1)}\text{(t)}\text{/}{R}^t,t\in {\mathbf{\text{N}}}_0\text{ or }{C}_{jk}=\lim B_{jk}^{(1)}\text{(}t\text{)}\text{/ }{\mathbf{\text{e}}}^{rt}\text{,}t\in {\mathbf{\text{R}}}^{+}.$

(ii) If {R = 1or r = 0 with N/t → ∞}and $D_k(t)\sim \{{\upsilon }_k\sqrt{u_{1}Bt}or{\upsilon }_k\sqrt{u_{1}bt}\}$, k = 1, …, d,then

$$\mathbf{\text{X}}(t;N)\stackrel{d}{\to }\eta =({\eta }_1,\dots ,{\eta }_d),$$ (33)

where η₁ = … = η_d a.s., η₁ ∊ N(0, 1).

(iii)If R > 1 or r > 0 and D_k(t) ~ {υ_kR^tor υ_ke^rt}, k = 1, …, d, then

$$\mathbf{\text{X}}(t;N)\stackrel{d}{\to }\zeta =({\zeta }_1,\dots ,{\zeta }_d) ,$$ (34)

where ζ₁ = … = ζ_d a.s., ζ₁ ∊ N (0, τ²) and τ² = IE{W² | Z(0) = δ₁} is defined in (30).

Proof

Using (31) one can introduce the multivariate characteristic functions

$${\Psi }_{t,N}(\mathrm{\alpha })=\text{IE}\{{\mathbf{\text{e}}}^{i\mathrm{\alpha }{\mathbf{\text{X}}}^T(t;N)}\}={[{\text{IE{}\mathbf{e}}^{i{{\displaystyle \sum }}_{k=1}^d{\mathrm{\alpha }}_k{\stackrel{\sim }{Z}}_k(t)/\sqrt{N}}|\mathbf{Z}\text{(0)}={\delta }_1\}]}^N,$$

where ${\stackrel{\sim }{Z}}_k(t)=[Z_k(t)-A_{1k}(t)]/D_k(t),k=1,\dots ,d.$

Further on we will use the following representation

$${\Psi }_{t,N}(\mathrm{\alpha })={\mathbf{\text{e}}}^{-i\sqrt{N}{{\displaystyle \sum }}_{k=1}^d{\mathrm{\alpha }}_k{A}_{1k}(t)/D_k(t)}{[{\text{IE{}\mathbf{e}}^{i{{\displaystyle \sum }}_{k=1}^d{\mathrm{\alpha }}_k{Z}_k(t)/D_k(t)\sqrt{N}}|\mathbf{Z}\text{(0)}={\delta }_1\}]}^N$$

$$={\mathbf{\text{e}}}^{-i\sqrt{N}{{\displaystyle \sum }}_{k=1}^d{\mathrm{\alpha }}_k{A}_{1k}(t)/D_k(t)}{F}^N(t;{\mathbf{\text{e}}}^{i{\mathrm{\alpha }}_1/D_1(t)\sqrt{N}},\dots ,{\mathbf{\text{e}}}^{i{\mathrm{\alpha }}_d/D_d(t)\sqrt{N}}) ,$$

where F (t; s) is introduced in (6).

Hence,

$$\log {\Psi }_{t,N}(\mathrm{\alpha })=-i\sqrt{N}{{\displaystyle \sum }}_{k=1}^d{\mathrm{\alpha }}_k{A}_{1k}(t)/D_k(t)+N\log F(t;{\mathbf{\text{e}}}^{i{\mathrm{\alpha }}_1/D_1(t)\sqrt{N}},\dots ,{\mathbf{\text{e}}}^{i{\mathrm{\alpha }}_d/D_d(t)\sqrt{N}}) .$$ (35)

(i) Note that in this case $D(t)\sqrt{N}\sim \sqrt{N{R}^t}(\text{or }\sqrt{N{\mathbf{\text{e}}}^{rt}})\text{ and }D(t)\sqrt{N}\to \infty .$ Therefore

$$\log {\Psi }_{t,N}(\mathrm{\alpha })\sim -i\sqrt{N}{{\displaystyle \sum }}_{k=1}^d{\mathrm{\alpha }}_k{A}_{1k}(t)/D(t)-N_{\phi t,N}(\mathrm{\alpha }) ,$$ (36)

where

$${\phi }_{t,N}(\mathrm{\alpha })=1-F(t;{\mathbf{\text{e}}}^{i{\mathrm{\alpha }}_1/D(t)\sqrt{N}},\dots ,{\mathbf{\text{e}}}^{i{\mathrm{\alpha }}_d/D(t)\sqrt{N}}) .$$ (37)

Hence developing (37) it is not difficult to see that

$$\begin{array}{cc}{\phi }_{t,N}(\mathrm{\alpha })\sim & -\sum _{k=1}^d(1-{\mathbf{\text{e}}}^{i{\mathrm{\alpha }}_k/D(t)\sqrt{N}})A_{1k}(t)\hfill \\ & -\frac{1}{2}\sum _{k,l=1}^d(1-{\mathbf{\text{e}}}^{i{\mathrm{\alpha }}_k/D(t)\sqrt{N}})(1-{\mathbf{\text{e}}}^{i{\mathrm{\alpha }}_l/D(t)\sqrt{N}})B_{jk}^{(1)}(t) .\hfill \end{array}$$

Then the following asymptotic result hold

$${\phi }_{t,N}(\mathrm{\alpha })\sim -(i/\sqrt{N})\sum _{k=1}^d{\mathrm{\alpha }}_k{A}_{1k}(t)/D(t)+\frac{1}{2N}\sum _{k,l=1}^d{\mathrm{\alpha }}_k{\mathrm{\alpha }}_l{B}_{kl}^{(1)}(t)/D^2(t) .$$ (38)

It is well-known that in the subcritical case

$$B_{jk}^{(i)}(t)\sim C_{jk}^{(i)}{R}^t,t\in {\mathbf{\text{N}}}_0;B_{jk}^{(i)}(t)\sim C_{jk}^{(i)}{\mathbf{\text{e}}}^{rt},t\in {\mathbf{\text{R}}}^{+}.$$ (39)

Further on we will use the notation $C_{jk}=C_{jk}^{(1)}$ and the corresponding matrix C= ‖C_jk‖.

Therefore from (36) – (39) one obtains log ${\Psi }_{t,N}(\mathrm{\alpha })\to -\frac{1}{2}{\displaystyle \sum _{k,l=1}^d{\mathrm{\alpha }}_k{\mathrm{\alpha }}_l{C}_{kl},}$ which proves (32) by the continuity theorem.

(ii) Note that in this case $D_k(t)\sqrt{N}\sim {\upsilon }_k\sqrt{u_1\mathit{BN}t}$ (or ${\upsilon }_k\sqrt{u_1\mathit{bNt}}$) and therefore (36) and (37) hold. Further on we will consider only the discrete-time case because the continuous-time case is similar as it was pointed out in the proof of Theorem 2. Then by (25) and (37) it is not difficult to obtain that

$$\begin{array}{cc}{\phi }_{t,N}(\mathrm{\alpha })\hfill & \sim \{u_1{\displaystyle \sum _{j=1}^d{\upsilon }_j}(1-{\mathbf{\text{e}}}^{i{\mathrm{\alpha }}_j/D_j(t)\sqrt{N}})\}/\{1+\frac{B}{2}t{\displaystyle \sum _{l=1}^d{\upsilon }_l}(1-{\mathbf{\text{e}}}^{i{\mathrm{\alpha }}_l/D_l(t)\sqrt{N}})\}\hfill \\ & \sim -(i{u}_1/\sqrt{N}){\displaystyle \sum _{j=1}^d{\upsilon }_j}{\mathrm{\alpha }}_j/D_j(t)/[1-(\mathit{iBt}/2\sqrt{N}){\displaystyle \sum _{l=1}^d{\upsilon }_l{\mathrm{\alpha }}_l/D_l(t)] .}\hfill \end{array}$$

From here using (36) and the fact that A_1k(t) ~ u₁υ_k one has

$$\log {\Psi }_{t,N}(\mathrm{\alpha })\sim -i{u}_1\sqrt{N}{\displaystyle \sum _{k=1}^d\frac{{\mathrm{\alpha }}_k{\upsilon }_k}{D_k(t)}+i{u}_1\sqrt{N}{\displaystyle \sum _{j=1}^d\frac{{\upsilon }_j{\mathrm{\alpha }}_j}{D_j(t)}/[1-i\frac{Bt}{2\sqrt{N}}{\displaystyle \sum _{l=1}^d\frac{{\upsilon }_l{\mathrm{\alpha }}_l}{D_l(t)}]=-(u_{1}Bt/2)\{{\displaystyle \sum _{k=1}^d{\frac{{\mathrm{\alpha }}_k{\upsilon }_k}{D_k(t)}\}}^2/[1-i\frac{Bt}{2\sqrt{N}}{\displaystyle \sum _{l=1}^d\frac{{\upsilon }_l{\mathrm{\alpha }}_l}{D_l(t)}] .}}}}}$$

Since $D_k(t)\sim {\upsilon }_k\sqrt{u_{1}Bt}$ then

$$\log {\Psi }_{t,N}(\mathrm{\alpha })\sim -\frac{1}{2}{\{{\displaystyle \sum _{k=1}^d}{\mathrm{\alpha }}_k\}}^2/[1-\frac{i\sqrt{B}}{2\sqrt{u_1}}\sqrt{\frac{t}{N}}{\displaystyle \sum _{l=1}^d{\mathrm{\alpha }}_l}] .$$

Finally ${\Psi }_{t,N}(\mathrm{\alpha })\to {\mathrm{e}}^{-\frac{1}{2}{\{{\sum }_{k=1}^d{\mathrm{\alpha }}_k\}}^2}$ which proves (33) by the continuity theorem.

(iii) In this case, using (30) and the references cited there, one has as $t\to \infty {\displaystyle \sum _{k=1}^d{\alpha }_k{W}_k(t)\to W{\displaystyle \sum _{k=1}^d{\alpha }_k{\upsilon }_k}},a.s.\text{ and }{L}_2.$

Therefore in the discrete-time case as t → ∞ the following convergence holds

$$F(t;{\mathbf{\text{e}}}^{i{\mathrm{\alpha }}_1/R^t},\dots ,{\mathbf{\text{e}}}^{i{\mathrm{\alpha }}_d/R^t})=\text{IE}\{{\mathbf{\text{e}}}^i{\sum }_{k=1}^d{\mathrm{\alpha }}_k{Z}_k(t)/R^t|\mathbf{Z}\text{(0)=}{\delta }_1\}$$

$$\to \mathrm{IE}\{{\mathrm{e}}^{\mathit{iW}{\sum }_{k=1}^d{\mathrm{\alpha }}_k{\upsilon }_k}|\mathbf{Z}\text{(0)=}{\delta }_1\}={\varphi }_1({{\displaystyle \sum }}_{k=1}^d{\mathrm{\alpha }}_k{\upsilon }_k).$$

Note that a similar relation is fulfilled also in the continuous-time case where only R^t has to be replaced by e^rt.

On the other hand, for the discrete-time processes the following equations hold

$${\varphi }_k(Rx)=F_k({\varphi }_1(x),\dots ,{\varphi }_d(x)),k=1,2,\dots ,d,$$ (40)

where ${\varphi }_k(x)=\mathrm{IE}\{{\mathbf{\text{e}}}^{\mathit{ixW}}|\mathbf{Z}\text{(0)=}{\delta }_k\},x\in {\mathbf{\text{R}}}^1,\text{and}{A}_k^{\ast }=IE\{W|\mathbf{\text{Z}}(0)={\delta }_k\}=u_k .$

Therefore as x → 0

$$1-{\varphi }_1(x)\sim -i{u}_{1}x+{\tau }^2{x}^2/2,$$ (41)

where τ² is defined in (34).

Note that in this case the relations (35) – (37) hold and by (37) and (41) one has

$${\phi }_{t,N}(\mathrm{\alpha })\sim 1-{\varphi }_1({\displaystyle \sum _{k=1}^d{\mathrm{\alpha }}_k/\sqrt{N}})\sim -i{u}_1{\displaystyle \sum _{k=1}^d{\mathrm{\alpha }}_k/\sqrt{N}}+({\tau }^2/2N){\{{\displaystyle \sum _{k=1}^d{\mathrm{\alpha }}_k}\}}^2.$$

Sine A_1k(t)/Dk(t) → u₁ then by (36) we obtain

$$\log {\Psi }_{t,N}(\mathrm{\alpha })\sim -i{u}_1\sqrt{N}{\displaystyle \sum _{k=1}^d{\mathrm{\alpha }}_k-N}[-i{u}_1{\displaystyle \sum _{k=1}^d{\mathrm{\alpha }}_k/\sqrt{N}+({\tau }^2/2N)}{\{{\displaystyle \sum _{k=1}^d{\mathrm{\alpha }}_k}\}}^2]=-({\tau }^2/2){\{{\displaystyle \sum _{k=1}^d{\mathrm{\alpha }}_k}\}}^2,$$

which proves (34) in the discrete-time case (applying the continuity theorem).

In the continuous-time case instead (40) the following system of differential equations holds

$$\frac{d}{dx}{\varphi }_k(x)=f_k({\varphi }_1(x),\dots ,{\varphi }_d(x))/rx,k=1,2,\dots ,d.$$ (42)

From (42) it is not difficult to obtain that $r{A}_k^{\ast }={\displaystyle \sum _{k=1}^d{a}_{kj}{A}_j^{\ast },k=1,2,\dots ,d}.$

Therefore by the Frobenius Theorem it follows that $A_k^{\ast }=C{u}_k$ for some constant C and it is not difficult to prove that C = 1. Indeed, since A_ik(t) ~ u_iυ_ke^rt then

$$\text{IE{}{W}_k(t)/{\upsilon }_k|\mathbf{Z}\text{(0)}={\delta }_1\}=A_{1k}(t)/{\upsilon }_k{\mathbf{\text{e}}}^{rt}\to u_1=A_1^{\ast }=\{W|\mathbf{Z}\text{(0)}={\delta }_1\}.$$

Hence (41) holds and the rest of the proof is similar as in the discrete-time case.

As it was pointed out in Remark 2, the relative frequencies Δ_i(t; N), i = 1, 2, …, d, introduced by (9), are strongly consistent and asymptotically unbiased estimators for p_i(t),i=1, 2, …,d, (defined by (11) and (12)) as N → ∞ for any fixed t. As one can see by the following theorem (proved in [32]), in this case the frequencies have also asymptotic multivariate normal distributions. Note that the theorem is valid for any kind of branching processes with discrete or continuous time (Markov or non Markov, reducible or irreducible) assuming only the usual independence of the individual evolutions.

Introduce the following notions:

$$\begin{array}{l}{\sigma }_i^2(t)=\text{Var{}{Z}_i(t) |\mathbf{Z}\text{(0)}={\delta }_1\},i=1,2,\dots ,d,\\ {\mathbf{\text{R}}}^{(d)}(t)=\parallel r_{ij}(t)\parallel ,r_{ij}(t)=\mathit{\text{Cor}}\{Z_i(t),Z_j(t)|\mathbf{Z}\text{(0)}={\delta }_1\},i,j=1,2,\dots ,d,\\ {\mathbf{\text{C}}}^{(d)}(t)=\parallel c_{ij}(t)\parallel ,c_{ii}(t)={\sigma }_i(t)(1-p_i(t)),c_{ij}(t)=-{\sigma }_i(t)p_j(t)\mathit{ for }i\ne j;i,j=1,2,\dots ,d.\end{array}$$

Theorem A

([32]). Let t be fixed and N → ∞. Then for the r.υ.

$$V_i(t;N)=[Z_i(t;N)-A_{1i}(t;N)]/[{\sigma }_i(t)\sqrt{N}]\text{ and}{W}_i(t;N)=M(t)\sqrt{N}[{\Delta }_i(t;N)-p_i(t)] ,$$

i = 1, 2, …, d, the following statements are valid:

(i) $(V_1(t;N),\dots ,V_d(t;N))\stackrel{d}{\to }(X_1(t),\dots ,X_d(t)),$

where the random vector X^(d)(t)=(X₁(t), …, X_d(t)) has a joint normal distribution with

$$\text{IE{}{X}_i(t)\}=0,\text{Var}\{X_i(t)\}=1\text{ and }\text{Cov}\{X_i(t),X_j(t)\}=r_{ij}(t).$$

(ii) $W_i(t;N)\stackrel{d}{\to }{Y}_i(t),$ where Y_i(t) is a normally distributed r.υ. with

$$\text{IE{}{Y}_i(t)\}=0\mathit{ and }\text{Var}\text{{}{Y}_i(t)\}={\displaystyle \sum _{j,k=1}^d{r}_{jk}(t)c_{ji}(t)c_{ki}(t),i=1,2,\dots ,d;}$$

(iii) For every k = 2, 3, …, d – 1

$$(W_1(t;N),\dots ,W_k(t;N))\stackrel{d}{\to }(Y_1(t), \dots ,Y_k(t)),$$

and Y^(k) (t) = (Y₁ (t), …, Y_k (t)) has a multivariate normal distribution with a covariance matrix which can be calculated as follows:

${\mathbf{\text{D}}}^{(k)}(t)=\parallel \text{Cov{}{Y}_i(t),Y_j(t)\}\parallel ={[{\mathbf{\text{C}}}_{d\times k}(t)]}^T{\mathbf{\text{R}}}^{(d)}(t){\mathbf{\text{C}}}_{d\times k}(t),$

where [C_d×k(t)]^T = ‖c_ji(t)‖, j = 1, 2, …, k; i = 1, 2, … d, is the corresponding transposed matrix of [k × d] dimensions.

Remark 3

While condition (10) implies that the fractions Δ_i(t; N), i = 1, 2, …, d, are linear dependent, there exist d − 1 joint normal distributions of lower dimensions that are asymptotically non-degenerate.

In a contrast of Theorem A the following results hold as both the number of the ancestors and the time increase to infinity.

Theorem 5

Assume N, t → ∞ and Condition (i) of Theorem 4. Then

(i) For every k = 1, 2, …, d

$${\Phi }_k(t;N)=u_{1}V\sqrt{ND(t)}[{\Delta }_k(t;N)-p_k(t)]\stackrel{d}{\to }{Y}_k,$$

where Y_k is a normally distributed r.υ. with IE{Y_k }= 0 and

$$\text{Var}\{Y_k\}=C_{kk}-2p{k}_{j=1}^d{C}_{kj}+p_{ki,j=1}^{2d}{C}_{ij}.$$

(ii) For every k = 2, 3, …, d − 1 and every subset (n₁, n_2,…, n_k) with nonrecurring elements from the set {1, 2, …, d} the following joint distributions hold

$({\Phi }_{n_1}(t;N),\dots ,{\Phi }_{n_k}(t;N))\stackrel{d}{\to }(Y_{n_1},\dots ,Y_{n_k}),$

where the random variables (Y_n1, …, Y_nk) have a multivariate normal distribution with

$$\mathit{Cov}(Y_{i,}{Y}_j)=C_{ij}-p_{\mathit{ik}=1}^d{C}_{\mathit{kj}}-p_{\mathit{jl}=1}^d{C}_{li}+p_i{p}_{\mathit{jk},l=1}^d{C}_{kl} .$$ (43)

Proof

Using (31) it is not difficult to obtain the following relations

$${\Phi }_k(t;N)=\frac{u_{1}VN{D}^2(t)}{U(t;N)}\{[1-P_k(t)]X_k(t;N)-p_k(t){\displaystyle \sum _{j\ne k}^d{X}_j(t;N)\},k=1,2,\dots ,d.}$$ (44)

Note first that from (17) and (22) one obtains $\frac{N{D}^2(t)u_{1}V}{U(t;N)}\to 1$ in probability. On the other hand, by (3), (11) and (12) it is not difficult to see that p_k(t) →p_k, k = 1, 2, …, d.Finally applying Theorem 4, (32), one has $X_j(t;N)\stackrel{d}{\to }{\xi }_j,$ where (ξ_1,…,ξ_d) have a multivariate normal distribution withzero means and a covariance matrix C =‖C_jk‖. Therefore by (44) it follows that

$${\Phi }_k(t;N)\stackrel{d}{\to }{Y}_k=(1-p_k){\xi }_k-p_k{\displaystyle \sum _{j\ne k}{\xi }_j}={\xi }_k-p_k{\displaystyle \sum _{j=1}^d{\xi }_j,k=1,2,\dots ,d,}$$ (45)

which obviously proves the statement (i).

The statement (ii) follows immediately by (44) and (45) applying Theorem 4, (i), where one have to check only that IE{Y_iY_j)} is well presented by (43).

Remark 4

Surprisingly in the critical and supercritical cases there are not analogs of Theorem 5. We will consider only the discrete-time case, because the continuous-time case is quite similar.

Indeed, under the conditions of Theorem 4, (ii), it is not difficult to see that

$$\sqrt{{N_u}_1/tB{p}_{\mathrm{k}}^2}[{\Delta }_k(t;N)-p_k(t)]\stackrel{d}{\to }{Y}_k=(1-p_k){\eta }_k-{\displaystyle \sum _{j\ne k}{p}_j}{\eta }_j, k=1,2,\dots ,d,$$

where η₁ = … = η_d a.s., η₁ ∊ N (0,1). Since ${\displaystyle \sum _{j=1}^d{p}_j=1}$ then Y_k = 0 a.s., k = 1, 2, …, d.

Similarly, under the conditions of Theorem 4, (iii),

$$\sqrt{N/p_k^2}[{\Delta }_k(t;N)-p_k(t)]\stackrel{d}{\to }{W}^{-1}{Y}_k\text{and}(U(t;N)/{\upsilon }_k{R}^t\sqrt{N})[{\Delta }_k(t;N)-p_k(t)]\stackrel{d}{\to }{Y}_k,$$

where

$$Y_k=(1-p_k){\varsigma }_k-{\displaystyle \sum _{j\ne k}^d{p}_j{\varsigma }_j ,}{\varsigma }_1=\dots ={\varsigma }_{d}a.s.,{\varsigma }_1\in N(0,{\tau }^2),{\tau }^2=\mathrm{IE}\{W^2 | Z(0)={\delta }_1\}.$$

Therefore Y_k = 0 a.s., k = 1, 2, …, d.

Remark 5

It is interesting to compare Theorem A with Theorem 4 and 5 (see also Remark 4). While Theorem A is valid for any kind of branching processes and does not depend of the criticality, then Theorem 4 and 5 are proved only in the Markov case and depend essentially by the usual trichotomy of the processes as well as by the relationship between N and t. Theorem 4 and 5 complete Theorem A in the case when t is large enough (but the obtained results are quite different).

6. Concluding Remarks

As it was pointed out in the Introduction and in Section 2, there are experimental situations in the cell biology where analyzing the relative frequencies, Δ_i(t; N), of cell types rather than the total cell counts Z_i(t; N), i = 1, 2, …, d, may be quite advantageous. Should this be the case, the property of asymptotic normality given by Theorem A ((ii) and (iii)) and Theorem 5 could be useful in developing the needed statistical inference of model parameters from experimental data. On the other hand, if the process can be observed directly then one can apply Theorem 4 or Theorem A, (i).

In particular, the following observation process is directly relevant to quantitative studies of proliferation, differentiation, and death of cells. Suppose that the process under study begins with $N={\displaystyle \sum _{k=1}^n{N}_k}$ cells of type T₁ and the values of N_k are all large, i.e., N₀ = min{N₁,N_2,…, N_n} → ∞ The descendants of the first N₁ ancestors are examined only once at time t₁ to determine the observations of Z_i(t₁; N₁) or Δ_i(t₁; N₁) = 1, 2, …, d, whereupon the observation process is discontinued (i.e. the cells under examination are destroyed). At the next moment t₂ $\ge$ t₁, the process Z_i(t₂; N₂) or the fractions Δ_i(t₂; N₂), i = 1, 2, …, d, related to the descendants of the second N₂ ancestors are observed, and so on. This procedure results in n independent observations of the form:

ζ_k = Z(t_k; N_k) = (Z₁(t_k; N_k), …, Z_d(t_k; N_k)) or ζ_k = Δ(t_k;N_k) = (Δ₁(t_k; N_k), …, Δ_d(t_k; N_k)), k = 1, 2, …, n; t₁ ≤ t₂ ≤ … ≤ t_n,

where each vector ζ_k is asymptotically normal in accordance with Theorems 4, 5 or A.

Denoting the corresponding contribution to the log-likelihood function by L_k(ζ_k; t_k, N_k), the overall log-likelihood is given by

$${\mathrm{\Lambda }}_n({\zeta }_{1,}{\zeta }_{2,}\dots ,{\zeta }_n)={\displaystyle \sum _{k=1}^n{L}_k({\zeta }_k;t_k,N_k).}$$ (46)

The log-likelihood (45) depends on the offspring parameters only, which are of primary interest in applications and especially in cell kinetics studies. Finally the parameters can be estimated from the data on the process or the relative frequencies by maximizing the log-likelihood (46). Some applications are given in [32].

Lamperti [22] obtained interesting limiting distributions for supercritical BGW processes with infinite offspring variance. It is an open problem to consider multitype processes with infinite variance when N and t tend to infinity.

Jagers [15] was probably the first to consider relative frequencies (proportions, fractions) of cells within the framework of multitype branching processes. He studied asymptotic (as t → ∞) properties of a reducible age-dependent branching process with two types of cells and proved convergence of their relative frequencies to non-random limits in mean square and almost surely on the non-extinction set. The usefulness of such frequencies in cell cycle analysis was further demonstrated by Mode [26] considering a four-type irreducible age-dependent branching process. Mode built his cell cycle analysis on a model of multitype positively regular age-dependent branching process. In the supercritical case, he proved that lim Δ_k(t) = δ_k a.s. as t → ∞, providing the population does not become extinct. It should be noted that the constants δ_k, k = 1, 2, …,d, depend only on the offspring characteristics. In fact, ${\delta }_k={\upsilon }_k/{\sum }_{j=1}^d{\upsilon }_j$, where ${\upsilon }_k={\eta }_k(1-G_k^{\ast }(\mathrm{\alpha })),\eta =({\eta }_{1,}{\eta }_{2,}\dots ,{\eta }_d)$ is a left eigenvector of the matrix $\mathbf{\text{H }}(\mathrm{\alpha })=\parallel G_k^{\ast }(\mathrm{\alpha })A\parallel$ with a Malthusian parameter α, while $G_k^{\ast }(\mathrm{\lambda })$ is the Laplace-Stieltjes transform of the life-span distribution G_k(t) for the type k(k= 1, 2, …, d). In his monograph, Mode [25] also considered the utility of fractions and reported a similar result for the BGW process.

The results obtained by Jagers and Mode for some age-dependent branching processes suggest that it would be interesting to investigate the asymptotic behavior of the fractions Δ(t; N) as well as the processes Z(t; N) in the non Markov case when both parameters N and t tend to infinity simultaneously. It is anticipated that such asymptotic properties will depend on a specific branching model and its reducibility. Another problem is the investigation (in the same situation) of the the most complicated models of branching processes such as considered in [2, 6, 34].