Large deviations for hitting times of some decreasing sets

In this paper 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 $b\rightarrow \infty$. 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 $(\frac{T_b}{b})$ as $b\rightarrow \infty$ with a large deviations principle concerning $(Z_t)$ which agrees with a generalized version of Mogulskii Theorem. The proof has some analogies with the proof presented in \cite{DW} for a similar result concerning nondecreasing univariate processes and their inverses with general scaling function.