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Large deviations for processes with independent increments. (English) Zbl 0769.60026

The large deviation principle (LDP) is shown for a family of probability measures \((P_ \lambda)\), \[ P_ \lambda(U)=P(\xi(tT)/r\in U),\quad \lambda=r^ 2/T, \] where \(\xi(t)\), \(t\geq 0\), is a stochastic process with stationary independent increments, \(U\subseteq D[0,1]\), \(r=r(T)\), \(r/T<\infty\), \(r/T^{1/2}\to\infty\) as \(T\to\infty\). The class of sets \(U\) is defined through the Skorokhod topology or the uniform-norm topology in \(D[0,1]\). Some theorems in [J. Lynch and J. Sethuraman, ibid. 15, 610-627 (1987; Zbl 0624.60045)] appear as corollaries of the LDP for \((P_ \lambda)\).
Reviewer: E.Pancheva (Sofia)

MSC:

60F10 Large deviations
60J99 Markov processes
60E07 Infinitely divisible distributions; stable distributions

Citations:

Zbl 0624.60045
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