Cooperative birth processes with linear or sublinear intensity
The asymptotic behavior of an increasing Markov process \(\{Z_ t,t\in {\mathbb{R}}^+\}\), \(Z_ t=(Z_ t(1),...,Z_ t(n))\in {\mathbb{N}}^ n\), is investigated. The increments are i.i.d. with finite expectation and covariances. Given \(Z_ t\), the time until the next jump is exponentially distributed with intensity \(h(Z_ t/| Z_ t|)f(Z_ t)\), where h and f are sufficiently smooth positive functions and \(| Z_ t| =\sum^{n}_{j=1}Z_ t(j)\). While a linear f results in asymptotically exponential growth, a suitable class of sublinear f leads to a growth at most that of a power. Covering both cases, analogues of the strong LLN, the CLT and LIL are obtained.
scientific article; zbMATH DE number 3870357
scientific article