Asymptotic behavior of the local times of a two-parameter random walk with finite variance

Abstract

In the paper one considers the asymptotic behavior for n→∞ of the number ϕ(m,n,j) of visits of a recurrent random walk$v_{\ell \kappa } = \mathop \Sigma \limits_{i = 1}^\ell \mathop \Sigma \limits_{j = 1}^k \xi ij$, where {ξij} ^∞_i,j=1 are independent, identically distributed, integer-valued random variables,Eξ₁₁=0,Eξ ²₁₁ =D<∞, at the point j up to the moment (m,n)m=m(n) Let$\hat t_n \left( {s, t, x} \right) = \left( {mn} \right)^{{\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 2$}}} \varphi \left( {\left[ {ms} \right], \left[ {nt} \right], \left[ {x\sqrt {mn} } \right]} \right)$ and let$\hat t\left( {s, t, x} \right)$ be the local time of the Brownian sheet W(s,t), EW (S,t)=Dst. One proves the weak convergence of the processes$\hat t\left( {s, t, x} \right)$ to the process$\hat t\left( {s, t, x} \right), \left( {s, t, x} \right) \in \left[ 0 \right., \left. \infty \right)^2 x R^1$ and this result is applied to the investigation of the limiting behavior of the functionals

$$\eta _n \left( {s, t} \right) = \sum\limits_{\ell = 1}^{\left[ {ms} \right]} {} \sum\limits_{\kappa = 1}^{\left[ {nt} \right]} {\sigma _n \left( {\ell , \kappa } \right)f_n \left( {v_{\ell \kappa } } \right)} ,\left( {s, t} \right) \in \left[ {0, T} \right]^2 ,$$

where σ_n(ℓ, κ) and f_n(v _ℓκ) are certain sequences of nonrandom functions.