The Chung Law of the Iterated Logarithm for Locally Square Integrable Martingales

Let X = （X_t,t≥ 0） be a locally square integrable martingale with X_o = 0. The predictable quadratic variation of X is＜X,X＞_t. Using the strong approximation result for continuous time semimartignales, we prove that if the jumps of X satisfy certain assumptions, the Chung law of the iterated logarithm for the locally square integrable martingale holds,that is \mathrmP\left(\liminf _t \rightarrow \infty \frac\sup _0 \leq s \leq t\left|X_s\right|\left(\langle X, X\rangle_t / \log \log (X, X\rangle_t\right)^1 / 2=\frac\pi\sqrt8\right)=1.