×

Limit theorems for switching processes. (English) Zbl 0821.60044

Butković, D. (ed.) et al., Functional analysis III. Proceedings of the postgraduate school and conference held at the Inter-University Center, Dubrovnik, Yugoslavia, October 29-November 12, 1989. Aarhus: Aarhus Universitet, Mat. Institut, Var. Publ. Ser., Aarhus Univ. 40, 235-261 (1992).
We introduce a special class of processes with discrete influence of chance – switching processes (SP). These processes can be described as two-component processes \((x(t), \zeta (t))\), \(t \geq 0\), with values in the space \((X, \mathbb{R}^ r)\) \((X\) is an arbitrary measurable space) such that there exists a sequence of moments \(\tau_ 1 < \tau_ 2 < \cdots\) so that on each interval \([\tau_ k, \tau_{k + 1})\), \(x(t) = x(\tau_ k)\) and the behaviour of \(\zeta (t)\) depends only on the value \((x (\tau_ k),\;\zeta (\tau_ k))\). The values \(\tau_ k\) are the moments of switchings and \(x(t)\) is the discrete switching component. These processes are given in terms of constructive characteristics and are suitable in describing systems which can vary spontaneously in moments which are random functionals of the previous trajectory (the moments of reaching some domain, of refusal etc.).
In the first part we consider the general limit theorems about the convergence of SP’s in the class of SP’s (the case when the number of switchings on each finite interval does not tend to infinity). In the second part we prove the principle of average and the convergence to diffusion processes for some special classes of SP’s when the number of switchings tends to infinity.
For the entire collection see [Zbl 0810.00021].

MSC:

60F99 Limit theorems in probability theory
60K30 Applications of queueing theory (congestion, allocation, storage, traffic, etc.)
PDFBibTeX XMLCite