The law of iterated logarithm for quadratic variations

V. A. Egorov

Zapiski Nauchnykh Seminarov POMI, Problems of the theory of probability distributions. Part X, Tome 158 (1987), pp. 72-80

Résumé

Let $\mathcal{P}_a$ be the class of those partitions $\pi$ of intervals $[0;T]$, such that $|t_i-t_{i-1}|>a$, where $a$ is a constant, $V(T,\mathcal{P}_a)=\underset{\pi\in\mathcal{P}_a}{\operatorname{sup}}\sum_i(w(t_i)-w(t_{i-1}))^2$. It is proved that for any $a$ $\lim V(T,\mathcal{P}_a)/2T\ln_2T=1$ a. s., where $\ln_2x=\ln\ln x$, if $\ln x\geq e$, $\ln_2x=1$, if $\ln x$.