Stopped Lévy processes with applications to first passage times
The main part of the paper is devoted to the limit theorems for the stopped homogeneous process with independent increments. Such results are well-known for the random walks; so this paper can be considered as an extension to the case of Lévy processes. One of such generalizations is stated below. Let \({\mathbf E} X(1)=0\), \({\mathbf E} X^2(1)=\sigma^2\) and \(\{\tau(t)\), \(t\geq 0\}\) be a family of stopping times such that \(\tau(t)/t \underset n\rightarrow\infty {\overset {p}\longrightarrow} \theta\in (0,\infty)\) as \(t\to \infty\). Then \(X(\tau(t))(\sigma^2\theta t)^{-1/2} \underset n\rightarrow\infty {\overset {d}\longrightarrow} N(0,1)\) as \(t\to \infty\). Similar generalizations have been made for the first passage time, a law of the iterated logarithm and for a stopped two-dimensional Lévy process. In conclusion the author considers the perturbed Lévy process \(Z(t)=X(t)+\xi(t)\), where it is supported that \(\xi(t)/t \underset n\rightarrow\infty {\overset \text{a.s.}\longrightarrow} 0\) as \(t\to \infty\). Central limit theorem and law of iterated logarithm have been stated for the random variable \(\nu(u)=\inf\{t: Z(t)>u\}\), \(u\geq 0\).
scientific article; zbMATH DE number 926962
scientific article