SOME RESULTS OF THE INCREMENT OF LOCAL TIME ON A RANDOM WALK

Let N（t） be the local time at zero （the number of returns to zero up to time t） of a reccurrent random walk. We obtain the following main theorem （1） （$\lim _{x \rightarrow \infty} \sup \frac{N(T)}{\sqrt{T \log \log T^T}}=\frac{\sqrt{2}}{\sigma}$ a.s （2）if $\lim _{T \rightarrow \infty} a_T / \log T=\infty$ then $\begin{aligned} & \lim _{T \rightarrow \infty} \sup _{a_x