Law of the iterated logarithm for \(\phi\)-mixing random variables
Let \(\{\xi_ i\}^{\infty}_{i=1}\) be a sequence of identically distributed random variables with zero expectation and finite variance. We put \(M^ b_ a=\sigma (\xi_ i,\quad a\leq i\leq b),\quad \phi (k)=\sup_{1\leq i<\infty}\sup_{A\in M^ i_ 1, B\in M^{\infty}_{i+k}}| P(B| A)-P(B)|,\quad S_ k=\sum^{k}_{i=1}\xi_ i,\) \(k=1,2,...\), \(\phi (0)=1\), \(\sigma_ n=DS_ n\). The basic result of this note is expressed in the following theorem. Let the following conditions be satisfied: 1) \(\sum^{\infty}_{k=1}\phi^{1/4}(k)k^ 2<\infty\); 2)\underbar{lim} \(\sigma\) \({}_ n/n>0\). Then the law of the iterated logarithm holds: \(\overline{\lim} S_ n/\sqrt{2\sigma_ n\ln \ln \sigma_ n}=1\) a.s.
scientific article; zbMATH DE number 3909432
scientific article