Limit theorems for switching processes. (Russian) Zbl 0638.60047

Anisimov, V. V.

Teor. Veroyatn. Mat. Stat., Kiev 37, 3-7 (1987).

Let \({\mathcal X}\) be a subset of a linear normed space, \(x^{(0)}\) a random variable in X, and \[ \{(\xi^{(\ell)}(t,u,x),\quad \chi^{(\ell)}(t,u,x)),\quad t\geq u\geq 0,\quad x\in {\mathcal X}\},\quad \ell \geq 0, \] be independent on \(x^{(0)}\) and mutually, families of random processes, which are measurable in a natural way in the parameters u, x. Here \(\chi\) (\(\cdot)\) are step processes the jumps of which take values in the space \({\mathcal X}\) and the trajectories of the processes \(\xi\) (\(\cdot)\) belong to the Skorokhod space \({\mathcal D}^ r_{\infty}\). Let \(\tau\) (u,x,\(\ell)\) be the moment of first jump of the process \(\chi^{(\ell)}(t,u,x)\), and \(\beta\) (u,x,\(\ell)\) be the value of this jump. Determine the recurrent sequences \(\zeta^{(\ell)}, x^{(\ell)}, \tau^{(\ell)}\), \(\ell \geq 0:\)
\(\tau\) \({}^{(0)}=0\), \(\tau^{(\ell +1)}=\tau (\tau^{(\ell)},x^{(\ell)},\ell)\), \(x^{(\ell +1)}=\beta (\tau^{(\ell)},x^{(\ell)},\ell),\)
\(\zeta\) \({}^{(0)}=0\), \(\zeta^{(\ell +1)}=\zeta^{(\ell)}+\xi^{(\ell)}(\tau^{(\ell +1)},\tau^{(\ell)},x^{(\ell)})\), \(\ell \geq 0.\)
Then the switching process \((x(t),\zeta (t))_{t>0}\) is determined in the following way: \[ x(t)=x^{(\ell)},\quad \zeta (t)=\zeta^{(\ell)}+\xi^{(\ell)}(t,\tau^{(\ell)},x^{(\ell)})\quad at\quad \tau^{(\ell)}\leq t<\tau^{(\ell +1)},\quad \ell \geq 0. \] A functional limit theorem about the convergence in the class of the switching processes and the theorem of the asymptotic enlargement of the switching processes are proved in the scheme of series.

Reviewer: V.V.Anisimov

MSC:

60F17	Functional limit theorems; invariance principles
60K99	Special processes

Keywords:

Skorokhod space; switching process; functional limit theorem; asymptotic enlargement of the switching processes