Large exceedances for multidimensional Lévy processes (Q1333382)

Let \(X: [0,\infty) \to \mathbb{R}^ d\) be a stochastic process with stationary independent increments and finite logarithmic moment generating function. Three asymptotic results concerning the increments of the normalized process \(Y_ t^ \varepsilon= X(t/\varepsilon)\) for \(\varepsilon\to 0\) are proved and discussed in this interesting paper: The first hitting time \(T_ \varepsilon\) of a rare set \(A\) is exponentially small; the duration of such a segment is of order \(1/\varepsilon\); its trajectory behaves as a deterministic straight line. The large deviations principle associated with the sample paths of \(Y^ \varepsilon\) is used to establish sufficient conditions for these asymptotic results, as soon as the set \(A\) satisfies certain geometrical conditions.