Large deviations for hitting times on some decreasing sets. (English) Zbl 1033.60036

Macci, Claudio

Bull. Belg. Math. Soc. - Simon Stevin 10, No. 3, 379-390 (2003).

Summary: We consider a suitable \(\mathbb{R}^d\)-valued process \((Z_t)\) and a suitable family of nonempty subsets \((A(b):b>0)\) of \(\mathbb{R}^d\) which, in some sense, decrease to empty set as \(b\to\infty\). In general let \(T_b\) be the first hitting time of \(A(b)\) for the process \((Z_t)\). The main result relates the large deviations principle of \((T_b/b)\) as \(b\to\infty\) with a large deviations principle concerning \((Z_t)\) which agrees with a generalized version of the Mogulskii theorem. The proof has some analogies with the proof presented by N. G. Duffield and W. Whitt [Ann. Appl. Probab. 8, 995–1026 (1998; Zbl 0939.60011)] for a similar result concerning nondecreasing univariate processes and their inverses with general scaling function.

MSC:

60F10

Large deviations

Keywords:

large deviations; first passage time; Mogulskii theorem; homogeneous function of degree 1

Citations:

Zbl 0939.60011