Journal of the
Korean Mathematical Society JKMS

We consider a general class of real-valued L\'evy processes $\{X(t),\ t\geq 0\}$, and obtain suitable large deviation results for the empiricals $L(t,A)$ defined by $t^{-1}\int^t_01_A(X(s)ds$ for $t>0$ and a Borel subset $A$ of ${\bf R}$. These results are used to obtain the asymptotic behavior of $P\{Z(t)\leq a\}$, where $Z(t) =\sup_{u\leq t}\vert x(u)\vert$, as $t\to\infty$, in terms of the rate function in the large deviation principle. A subclass of these processes is the Feller class: there exist nonrandom functions $b(t)$ and $a(t) > 0$ such that $\{(X(t)-b(t))/a(t):t>0\}$ is stochastically compact, i.e., each sequence has a weakly convergent subsequence with a nondegenerate limit. The stable processes are in this class, but it is much larger. We consider processes in this class for which $b(t)$ may be taken to be zero. For any $t>0$, we consider the renormalized process $\{X(v\psi (t))/a(\psi (t)),\ v\geq 0\}$, where $\psi (t) = t(\log\log t)^{-1}$, and obtain large deviation probability estimates for $L_t(A):= (\log\log t)^{-1}\int^{\log\log t}_0 1_A(X(v\psi (t))/a(\psi (t)))dv$. It turns out that the upper and lower bounds are sharp and depend on the entire compact set of limit laws of $\{X(t)/a(t)\}$. The results extend to random walks in the Feller class as well. Earlier results of this nature were obtained by Donsker and Varadhan for symmetric stable processes and by Jain for random walks in the domain of attraction of a stable law.

Keywords: large derivations of empiricals, L\'evy processes, random walks, Feller distributions, stochastic compactness

MSC numbers: 60E07, 60F10, 60G17, 60G50