Stopped two-dimensional perturbed random walks and Lévy processes
The author presents some limit theorems for stopped and perturbed two-dimensional random walks and Lévy processes. In particular, a strong limit theorem for the process(es) and a central limit theorem for the overshoots are established. The results are subsequently specialized to processes of the type \(Z^{(i)}_n= ng^{(i)}(n^{-1}(X^{(i)}_1+\cdots+ X^{(i)}_n))\), \(i= 1,2\), with i.i.d. sequences \(\{X^{(i)}_n\}_{n\in \mathbb{N}}\) of integrable random variables and certain real functions \(g_1\), \(g_2\). An application to repeated significance tests in two-parameter exponential families is given. The paper contains only rough sketches of some proofs. For most of the material the reader is being referred to (suitably adapted versions of the) proofs in various previously published papers by the author and co-authors.
scientific article; zbMATH DE number 1092021
scientific article