Limit theorems for evolving accumulation processes. (English. Russian original) Zbl 0746.60019

Anisimov, V. V.

Theory Probab. Math. Stat. 43, 5-11 (1991); translation from Teor. Veroyatn. Mat. Stat., Kiev 43, 6-13 (1990).

For \(n=1,2,\dots\) let \(\nu_ n(t),\;t\geq 0\), be a flow of events with \(t_{nk}\) as their moments of occurrence, and let \(\xi_{nk}(t),\;t\geq 0\), be independent families of \(R^ m\)-valued random processes whose sample paths belong to the space \(D_{[0,\infty)}(R^ m)\) and their distributions are independent of \(k\). The author investigates the processes \[ S_ n(t) = \sum_{k=1}^{\nu_ n(t)} \xi_{nk} (t- t_{nk}),\quad t\geq 0,\quad S_ n(0)=0, \quad n=1,2,\dots \] Under some conditions it is proved that the finite-dimensional distributions of \(S_ n(t)\) weakly converge to those of a random process.

Reviewer: V.M.Kruglov (Moskva)

MSC:

60F05	Central limit and other weak theorems
60G55	Point processes (e.g., Poisson, Cox, Hawkes processes)

Keywords:

weak convergence; finite-dimensional distributions