Occupation times of subcritical branching immigration systems with Markov motion, CLT and deviation principles
The author studies two closely related stochastic models. The first is a subcritical branching particle system with immigration. The particles move independently in \(\mathbb{R}^d\) according to a time-homogeneous Markov family. The lifetime of each particle is exponential; when a particle dies it has either two offspring particles with probability \(q\) or none with probability \(1-q\), where \(q < 1/2\). New particles immigrate according to a homogeneous Poisson random field in time and space.NEWLINENEWLINEThe second model is a limiting case of the first, a measure-valued time-homogeneous Markov process, that is, a superprocess. Let \((N_t)_{t \geq 0}\) denote either of these two processes; the main objective of the paper is to study the rescaled occupation time process NEWLINE\[NEWLINE Y_T(t) := \frac{1}{F_T} \int_0^{Tt} N_s\, ds, \quad t \geq 0, NEWLINE\]NEWLINE and its fluctuation NEWLINE\[NEWLINE X_T(t) := \frac{1}{F_T} \int_0^{Tt} (N_s - \operatorname{E}N_s)\, ds, \quad t \geq 0, NEWLINE\]NEWLINE where \(F_T\) denotes a suitable deterministic norming which may vary from case to case.NEWLINENEWLINEUnder certain assumptions, the paper gives functional central limit theorems, large and moderate deviation theorems for the processes \(X_T\) and \(Y_T\), with both models. A more detailed version of the paper can be found in the extended version [\url{arXiv:0911.0777}].
scientific article; zbMATH DE number 6044890
scientific article