Stability and attraction to normality for Lévy processes at zero and at infinity (Q1866065)

For a Lévy process \(X_t\) the weak and almost sure behaviour is considered for \(X_t/b(t)\to 1\) and \(X_t/t \to\delta\) as \(t\downarrow 0\) or \(t\to \infty\). Necessary and sufficient conditions are obtained expressed in terms of characteristics directly available from the characteristic functions of the process. The convergence of these ratios to infinity and the weak convergence to normal distributions are considered as well.