An asymptotic behaviour of local times of a recurrent random walk with finite variance

The paper deals with the asymptotic behaviour (as $n\to\infty$) of the number $\varphi(n,r)$ of times the recurrent random walk $\nu_k$ hits the point $r$ till time $n$. We prove that if the random walk has a finite variance then the processes
$$ t_n(t,x)=n^{-1/2}\varphi([nt],[x\sqrt n]),\qquad(t,x)\in[0,\infty)\times\mathbf R^1 $$
(where $[a]$ is the integer part of $a$), converge weakly to the process $\mathbf t(t,x)$ – the Brownian local time at the point $x$ after time $t$. This result is applied to the investigation of a limit behaviour of a number of processes generated by a recurrent random walk $\nu_k$.