A law of the iterated logarithm for processes with independent increments

Abstract

By using the Ito's calculus, a law of the iterated logarithm is established for the processes with independent increments (PII). LetX = {X _t,t ≥ 0} be a PII withEX _t = 0,V(t) =EX ^t₂ < ∞ and lim _t→∞ V(t) = ∞. If one of the following conditions is satisfied,

$$\begin{gathered} E\sup _t \left( {\frac{{\Delta X_t }}{{g(t)}}} \right)^2< \infty for some \varepsilon (t) = o\left( {\left( {\frac{{V(t)}}{{LLg(V(t))}}} \right)^{1/2} } \right),where \Delta X_t = \hfill \\ X_t - X_{t - } and LLg (x) = log(x V e^e )). \hfill \\ \end{gathered} $$

((1))

(2) Suppose the Levy's measure ofX may be written asV(dt, ds) =F _t(dx)dV (t) and there is aσ-finite measureG such thatf _{|y| ≥x} F _t(dy) ≤a φ_|y|≥x G(dy) and φy ² G(dy) < ∞, then

$$P\left( {\overline {\mathop {\lim }\limits_{t \to \infty } } \frac{{|X_t |}}{{\sqrt {2V(t)LLg} (V(t))}} = 1} \right) = 1.$$

((2))