MATHEMATICS

V. P. CHISTYAKOV

TRANSIENT PHENOMENA IN BRANCHING PROCESSES WITH \(n\) TYPES OF PARTICLES

(Presented by Academician A. N. Kolmogorov, December 3, 1959)

§ 1. Consider the branching random process defined in \((^{1})\), with \(n\) types of particles, homogeneous in time. Let, during a time interval \(\Delta t \to 0\), a particle of type \(T_k\) turn into a set of particles \(\omega=(\omega_1,\ldots,\omega_n)\) of types \(T_1,\ldots,T_n\) with probabilities \(\delta_k^\omega+p_k^\omega\Delta t+o(\Delta t)\), where \(\delta_k^\omega=1\) if \(\omega_k=1\), \(\omega_i=0\) \((i\ne k)\); \(\delta_k^\omega=0\) in all other cases. The densities \(p_k^\omega<0\) if \(\delta_k^\omega=1\); \(p_k^\omega\ge 0\) if \(\delta_k^\omega=0\); \(\sum_\omega p_k^\omega=0\). Denote by \(\mu_{kj}(t)\) the number of particles of type \(T_j\) obtained from one particle of type \(T_k\) during time \(t\). The generating functions \(F_k(t,x_1,\ldots,x_n)\) of the probabilities \(P_k^\omega(t)=\mathbf{P}\{\mu_{kj}(t)=\omega_j,\ j=1,\ldots,n\}\) satisfy the system of equations \((^{1})\)

\[ \frac{dF_k}{dt}=f_k(F_1,\ldots,F_n), \qquad k=1,\ldots,n \tag{1} \]

(where \(f_k(x_1,\ldots,x_n)=\sum_\omega p_k^\omega x_1^{\omega_1}\cdots x_n^{\omega_n}\)) and the boundary conditions

\[ F_k(0,x_1,\ldots,x_n)=x_k \qquad (k=1,\ldots,n). \]

Denote the factorial moments of the functions \(f_k(x_1,\ldots,x_n)\) by
\[ a_{ij}=\partial f_i/\partial x_j\big|_{x=1}, \quad b_{ij}^{(k)}=\partial^2 f_k/\partial x_i\partial x_j\big|_{x=1}, \quad c_{i,j,l}^{(k)}=\partial^3 f_k/\partial x_i\partial x_j\partial x_l\big|_{x=1}. \]
For processes with an indecomposable matrix \(a=\|a_{ij}\|\), as \(t\to\infty\),

\[ M\mu_{kj}(t)\sim u_k v_j e^{\lambda t} \qquad (k,j=1,\ldots,n), \]

where \(\lambda\) is the characteristic number with largest real part of the matrix \(a\); \(u=(u_1,\ldots,u_n)\) and \(v=(v_1,\ldots,v_n)\) are eigenvectors of the number \(\lambda\) for the matrix \(a\) and its transpose, respectively, with

\[ \sum_{k=1}^{n} v_k^2=1,\qquad \sum_{k=1}^{n} u_k v_k=1, \]

\[ u_k>0,\quad v_k>0 \qquad (k=1,\ldots,n). \]

Put
\[ b=\sum_{i,j,k=1}^{n} b_{ij}^{(k)} v_k u_i u_j. \]
For processes with an indecomposable matrix \(a\) and \(0<b<\infty\), the probabilities of continuation of the process (begun with one particle of type \(T_i\))
\[ Q_i(t)=\mathbf{P}\left\{\sum_{j=1}^{n}\mu_{ij}(t)>0\right\} \]
as \(t\to\infty\) satisfy the relations \((^{1,2,6})\)

\[ Q_i(t)\sim \begin{cases} K_i e^{\lambda t}, & \text{if } \lambda<0,\\ 2u_i/bt, & \text{if } \lambda=0,\\ 1-P_i, & \text{if } \lambda>0. \end{cases} \]

where \(K_i>0\) are constants, \(P_i\) satisfy the equations \(f_k(P_1,\ldots,P_n)=0\) \((k=1,\ldots,n)\), and the conditional distribution laws \((^2,^3,^6)\)

\[ S_k(t,\lambda,y)=\mathbf P\left\{\frac{\mu_{kj}(t)} {M\{\mu_{kj}(t)\mid \mu_k(t)>0\}}<y_j,\ j=1,\ldots,n\mid \mu_k(t)>0\right\} \tag{2} \]

(where \(y=(y_1,\ldots,y_n)\) and \(\mu_k(t)=\sum_{j=1}^n \mu_{kj}(t)\)) converge weakly to the distributions \(S_k(y)\).

For \(\lambda<0,\ \lambda>0\), the distributions \(S_k(y)\) depend essentially on the form of the functions \(f_k(x)\) \((k=1,\ldots,n)\); moreover, in the first case the distributions are discrete, and in the second continuous. For \(\lambda=0\), the distributions \(S_k(y)\) are exponential, with characteristic functions

\[ \varphi_k(\tau_1,\ldots,\tau_n)=\frac{1}{1-i(\tau_1+\cdots+\tau_n)} . \tag{3} \]

Thus, a branching process with \(\lambda=0\) separates two mutually distinct types of branching processes. We shall call transition phenomena those phenomena that arise as \(\lambda\to0\).

In the present note we shall give limit theorems for \(t\to\infty,\ \lambda\to0\), analogous to the theorems in \((^4)\), obtained for processes with one particle type. The results stated above for processes with \(\lambda=0\) can be obtained as consequences of Theorems 1 and 2.

§ 2. Let \(\mathfrak A\) denote a closed compact set (in the sense of convergence by elements) of indecomposable matrices of the form \(a\) with \(\lambda\) close to 0. We shall consider branching processes from the class \(K(\mathfrak A,\delta,B,c)\). A process belongs to \(K(\{f_k\}\in K)\) if \(a\in\mathfrak A,\ 0<\delta<b<B<\infty,\ c_{ijl}^{(k)}\leq c<\infty\ (c\geq0)\). Define the function

\[ k(t,x,\lambda)= \frac{e^{\lambda t}\sum_{j=1}^n v_j(1-x_j)} {1+\dfrac b2\,g(\lambda,t)\sum_{j=1}^n v_j(1-x_j)}, \]

where \(g(\lambda,t)=t\) for \(\lambda=0\) and \(g(\lambda,t)=\dfrac{e^{\lambda t}-1}{\lambda}\) for \(\lambda\neq0\).

Theorem 1.

\[ 1-F_j(t,x)=u_j k(t,x,\lambda)[1+\eta_j(t,x,\lambda)], \]

where \(\eta_j(t,x,\lambda)\to0\) as \(t\to\infty,\ \lambda\to0\), uniformly in \(\{f_k\}\in K,\ 0\leq x_k\leq1\) \((k=1,\ldots,n)\).

Corollary. As \(t\to\infty,\ \lambda\to0\), uniformly in \(\{f_k\}\in K\),

\[ Q_j(t)\sim \frac{2u_j e^{\lambda t}}{b g(\lambda,t)} . \]

Theorem 2. As \(t\to\infty,\ \lambda\to0\), uniformly in \(\{f_k\}\in K\),

\[ \max_y |S_k(t,\lambda,x)-S(y)|\to0 \]

(where \(S(y)\) is defined by (3)).

We shall briefly outline the proofs of the formulated theorems.

Proof of Theorem 1. First we establish that

\[ Q(t)=\sum_{k=1}^n Q_k(t)v_k\to0 \]

as \(t\to\infty,\ \lambda\to0\), uniformly in \(\{f_k\}\in K\). Next put

\[ r_k(t,x)=\sum_{j=1}^n \pi_{kj}R_j(t,x), \]

where \(k=1,\ldots,n,\ R_j(t,x)=1-F_j(t,x)\). Choose the matrix \(\Pi=\|\pi_{kj}\|\) so that: 1) the matrix \(\Pi a\Pi^{-1}\) has Jorda-

new form; 2) the first row of \(P\) was proportional to \(v\); 3) the ratios \(r_k(t,x)/r_1(t,x)\) \((k\ne 1)\) were sufficiently small uniformly in \(t\), \(0\le x_j\le 1\) \((j=1,\ldots,n)\), \(\{f_k\}\in K\) (this is possible, since
\[ |r_k|\le \sum_{j=1}^{n}|\pi_{kj}|R_j(t,x)\le \frac{\max_j|\pi_{kj}|}{\min_j v_j}\sum_{j=1}^{n} v_j R_j(t,x), \]
and \(\min_j v_j>\delta_1>0\) for any \(a\in \mathfrak A\)).

To prove the theorem it is enough to show that, as \(t\to\infty\), \(\lambda\to0\), uniformly for \(0\le x_k\le 1\) \((k=1,\ldots,n)\), \(\{f_k\}\in K\),
\[ \frac{r_k(t,x)}{r_1(t,x)}\longrightarrow 0 \qquad (k\ne 1); \tag{4} \]
\[ r_1(t,x)\sim k(t,x,\lambda), \tag{5} \]
since it follows from (4) that \(R_j(t,x)\sim u_j r_1(t,x)\).

The proof of (5) is carried out analogously to the proof of theorem 1 in \((^4)\). Relations (4) can be proved with the aid of the inequality
\[ d\rho^2/dt<-\delta_0\rho^2+CQ(t)\quad \left(\delta_0>0,\ C>0\text{ are constants},\ \rho^2=\sum_{k=2}^{n}|r_k|^2\bigg/\sum_{k=2}^{n}|r_k|^2\right), \]
which can be obtained from (1) by applying transformations analogous to the transformations in the proof of theorem 2, p. 121 in \((^5)\), if one takes into account that \(r_k(t,x)/r_1(t,x)\) \((k\ne 1)\) are uniformly small.

Proof of theorem 2. If theorem 2 is false, then there exist \(\varepsilon>0\) and a sequence \(S_k(t^{(m)},\lambda^{(m)},y)\) such that
\[ \max_y |S_k(t^{(m)},\lambda^{(m)},y)-S(y)|>\varepsilon \]
and \(t^{(m)}\to\infty\), \(\lambda^{(m)}\to0\) as \(m\to\infty\). By theorem 1 it is not difficult to compute that the Laplace transforms of the distributions \(S^k(t^{(m)},\lambda^{(m)},y)\) as \(m\to\infty\) converge to
\[ \frac{1}{1+s_1+\cdots+s_n} \]
(\(s_j\ge0,\ j=1,\ldots,n\)). Since \(S(y)\) is continuous, it follows from this that
\[ \max_y |S_k(t^{(m)},\lambda^{(m)},y)-S(y)|_{m\to\infty}\longrightarrow 0. \]

The contradiction obtained proves the theorem.

§ 3. For processes with \(n\) types of particles, starting with a set \((m_1,\ldots,m_n)\) of particles of types \(T_1,\ldots,T_n\), as \(t\to\infty\), \(\lambda\to0\), \(m_j\to\infty\) \((j=1,\ldots,n)\), by means of theorem 1 and lemma 7 from \((^4)\) one can obtain analogues of theorems 3 and 4 from \((^4)\). The limiting distributions will be concentrated on a one-dimensional straight line, and on this line they will coincide with the limiting distributions of theorems 3 and 4 from \((^4)\).

The author expresses gratitude to B. A. Sevastyanov for formulating the problem and for the suggestions received in the course of its solution.

Moscow State University
named after M. V. Lomonosov

Received
27 XI 1959

REFERENCES

\(^1\) B. A. Sevastyanov, Uspekhi Mat. Nauk, 6, No. 6 (1951).
\(^2\) M. Irzhina, Czechoslovak Mathematical Journal, 7 (82), No. 1 (1957).
\(^3\) T. E. Harris, Proc. Second Berkeley Symposium on Mathematical Statistic and Probability, 1951.
\(^4\) B. A. Sevastyanov, Theory of Probability and Its Applications, 4, issue 2 (1959).
\(^5\) V. V. Nemytskii, V. V. Stepanov, Qualitative Theory of Differential Equations, 1947.
\(^6\) V. P. Chistyakov, Summary of a report, Theory of Probability and Its Applications, 4, issue 4 (1959).