Structural, continuity, and asymptotic properties of a branching particle system
Let \(W(\delta,q)\) be a Pólya-Aeppli distributed random variable with parameters \(\delta\in\mathbb{R}+\) and \(q\in (0,1)\), i.e., \(\log\,{\mathbf E}\exp\{sW(\delta, q)\}= \delta(e^s- 1)/(1/(1- q)- e^s)\) for \(s<-\log(1-q)\). Note that the range of this r.v. is \(\mathbb{Z}+\) and that the random sum of \(W(\delta,q)\) i.i.d. random variables, which follow a common zero-modified geometric law, is again Pólya-Aeppli distributed. These properties are particularly useful in the context of classical branching processes. Consider a Markov branching process \({{\mathfrak L}^{(\eta)}}_{\delta, t}\) started with the number \(W(\delta, \delta/(\delta+\eta))\) (with mean \(\eta\)) of particles at time \(t= 0\). Let the particles move on \(\mathbb{R}^d\), each undergoing critical binary branching after an exponentially distributed lifetime with mean \(1/\eta\). The particles are identical, their motions, lifetimes and branching independent of each other. Let \({\widetilde T^{(\eta)}}_\delta(t)\) denote the of particle alive at time \(t= 0\). Then, for each \(t> 0\), \({\widetilde L^{(\eta)}}_\delta(t)\) is distributed as \(W(2\delta/(2+\delta t), 2/(2+2\eta/\delta t))\). Now let the random measure \({{\mathfrak L}^{(\eta)}}_{\delta,0}\) at time \(t_0= 0\) represent the current state of a process that originated sometime in the past. Assume that the evolution mechanism from past to present was the same as it is for the future and that the number of particles alive at an admissible instant \(s< 0\) is Pólya-Aeppli distributed with parameters \(\Delta_s\) and \(Q_s\), say. Then the precise lower bound for the admissible times is \(-2/\delta\), and the parameter values at \(s\in(-2/\delta,0)\) are \(\Delta_s= 2\delta/(2+s\delta)\) and \(Q_s= \Delta_s/(\eta+\Delta_s)\). The process \(\{{\widetilde L^{(\eta)}}_\delta(s); s> -\delta/2\}\) must have originated from a Poisson field with mean \(1/\eta\) at \(s= -2/\delta\), and it has a realization continuous in mean square. Explicit expressions for the coefficients of \(1/\eta\) and \(1/\eta^2\) in the expansion of \({\mathbf P}({\widetilde L^{(\eta(}}_\delta(t)/\eta= x)-\chi(\{x= 0\})e^{-2\delta/(2+\delta t)}\) into a power series in \(1/\eta\) are obtained.
continuity of stochastic process in mean square
scientific article; zbMATH DE number 5626712
scientific article