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A law of the iterated logarithm for processes with independent increments

Authors:

Jiagang Wang

Institution:

1. East China University of Science & Technology, 200237, Shanghai, China

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 ₂ ^t <> infin

and lim_t V(t) = infin

. If one of the following conditions is satisfied,

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

Keywords:	Law of the iterated logarithm process with independent increments locally square integrable martingale Ito's calculus
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