On Chung's Law of the Iterated Logarithm for Some Stochastic Integrals

Abstract

We prove that there exists a constant $a(A) \in (0, \infty)$ such that $\lim \inf_{t \rightarrow \infty} (\log \log t/t)\sup_{0 \leq s \leq t}|\int^s_0\langle AW_u, dW_u\rangle | = a(A)$ with probability 1, where $A$ is a skew-symmetric $d \times d$ matrix, $A \neq 0$, and $\{W_t\}_{t\geq 0}$ is a $d$-dimensional Wiener process.