engleski

Asymptotic analysis of subcritical branching processes with immigration

When considering stationary multivariate regularly varying time series, it is useful to observe that, conditionally on the event that the norm of the present value exceeds a given threshold $x$, the whole sequence normalized by $x$ has a limiting distribution as $x \to \infty$. That limit is called the tail sequence. Provided that time series satisfies some weak dependence conditions, its extremal behavior can be elegantly characterized using the notion of the tail process and the theory of point processes. However, except in a few simple cases, establishing such conditions and determining exact distribution of the tail process in the multivariate setting remains a technically challenging task. Here we study a class of models where we show a somewhat different route to asymptotic analysis. The presentation is motivated by the study of conditional least squares estimator of the mean number of progeny in the branching process with heavy tailed immigration. We also exhibit the rate of convergence and precise asymptotic distribution of the estimator.

branching processes with immigration ; point processes ; limit theorems

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