Limit Theorems for Branching Stochastic Processes of Special Form

The following scheme of generating particles is considered. Each particle existing at a given time turns into k particles with the probability $\delta _{k1} + p_k \Delta t + o(\Delta t)$ in time $\Delta t \to 0$ independently of its age and origin and the history of the other particles. Moreover, k particles arise with the probability $\delta _{k0} + q_k \Delta t + o(\Delta t)$ in time $\Delta t \to 0$ no matter how many other particles may be present. Here $\sum\nolimits_{k = 1}^\infty {p_k } = 0$, $\sum\nolimits_{k = 1}^\infty {q_k } =0$, $\delta _{ii} = 1$, $\delta _{ii} = 0$ if $i \ne j$. We define the probability-generating functions by \[ f(x) = \sum\limits_{k = 0}^\infty {p_{k^{x ^k } } } ,\qquad g(x) = \sum\limits_{k = 0}^\infty {p_{k^{x ^k } } } \] and denote factorial moments by \[ f'(1) = a_1,\quad f''(1) = b_1 , \quad f'''(1) = c_1 , \quad g'(1) = a_2 , \quad g''(1) = b_2 . \] Let $\mu _t $ be the number of particles at time t if $\mu _0 = 0$. The following limit theorems are proved. Theorem 1.If$a_1 < 0, a_2 < \infty $, then the limits\[ \mathop {\lim }\limits_{t \to \infty } {\bf P}\{ \mu _t = k\} = P_k ,\quad k = 0,1,2, \cdots , \]exist, and the probability-generating function\[ F(x) = \sum\limits_{k = 0}^\infty {p_{k^{x^k } } } \]is defined by (15). Theorem 2.If$a_1 = 0$and$b_1 > 0,c_1 ,a_2 > 0,b_2 $are finite, then the limit relation (19) holds true. Theorem 3.If$a_1 = 0,b_1 ,a_2 ,b_2 $are finite, then\[ \mathop {\lim }\limits_{t \to \infty } {\bf P}\{ \mu _t/e^{a_1 t} < y\} = S(y) \]exists and the distribution function$S(y)$has a characteristic function defined by equations (28) and (27).

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