Geodesic
An asymptotic behaviour of local times of a~recurrent random walk with finite variance
Teoriâ veroâtnostej i ee primeneniâ, Tome 26 (1981) no. 4, pp. 769-783

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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 $\nu_k$. 
@article{TVP_1981_26_4_a7,
     author = {A. N. Borodin},
     title = {An asymptotic behaviour of local times of a~recurrent random walk with finite variance},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {769--783},
     publisher = {mathdoc},
     volume = {26},
     number = {4},
     year = {1981},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1981_26_4_a7/}
}
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A. N. Borodin. An asymptotic behaviour of local times of a~recurrent random walk with finite variance. Teoriâ veroâtnostej i ee primeneniâ, Tome 26 (1981) no. 4, pp. 769-783. http://geodesic.mathdoc.fr/item/TVP_1981_26_4_a7/