An asymptotic behaviour of local time of two-parameter random walk with finite variance

Let $\hat t(s, t, x)$ be the local time of the Brownian sheet $w(s, t)$, $\mathbf Ew^2(s,t)=Dst$, $\hat t_n(s, t, x)=(mn)^{-1/2}\varphi([ms], [nt], [x\sqrt{mn}])$ being the number of times the recurrent random walk $\nu_{lk}=\sum_{i=1}^l\sum_{j=1}^k\xi_{ij}$ hits the point $j$ till $(m, n)$, $m=m(n)$, where $\{\xi_{ij}\}$ are i. i. d. integral-valued r. v., $\mathbf E\xi_{11}=0$, $\mathbf E\xi_{11}^2=D<\infty$. The weak convergence $\hat t_n\to\hat t$ is proved and applications to investigation of the behaviour of functionals
$$ \eta_n(s, t)=\sum\sum\sigma_n(l, k)f_n(\nu_{lk}),\quad(s, t)\in[0, T]^2 $$
are given ($\sigma_n$, $f_n$ are nonrandom functions).