Asymptotic properties for Cauchy's principal values of Brownian and random walk local time

We present a strong approximation result between the Cauchy's principal values of Brownian local time and general random walk local time, and obtain the law of the iterated logarithm of Cauchy's principal values of random walk local time. Furthermore, we prove some precise asymptotics in the law of the iterated logarithm of Cauchy's principal values of random walk local time.