Two Chung-type and Hirsch-type laws are established to describe the liminf asymptotic behaviours of the Cauchy's principal values related to Brownian local times. These results are generalized to a class of Brownian additive functionals.
"The Laws of Chung and Hirsch for Cauchy's Principal Values Related to Brownian Local Times." Electron. J. Probab. 5 1 - 16, 2000. https://doi.org/10.1214/EJP.v5-66