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A local limit theorem for nonhomogeneous random walk on a lattice
E. A. Zhizhinaa,
R. A. Minlosb a Moscow Power Engineering Institute
b Institute for Information Transmission Problems, Russian Academy of Sciences
Abstract:
In this paper, we study the walk of a particle on the
$\nu$-dimensional lattice
$\mathbf{Z}^\nu$,
$\nu=1,2,3$, of which the one-step transition probabilities
$\mathbf{Pr}(y\to x)$ differ from those of the homogeneous symmetric walk only in a finite neighborhood of the point
$x=0$. For such a walk, the main term (having the order
$O(1/t^{\nu/2})$) of the asymptotics of the probability
$\mathbf{Pr}(x_t=x\mid x_0=y)$ as
$t\to\infty$ is studied,
$x,y\in\mathbf{Z}^\nu$,
$x_t$ being the position of the particle at time
$t$. It turns out that, for
$\nu=2,3$, this main term of the asymptotics differs from the corresponding term of the asymptotics for the homogeneous walk (which has a usual Gaussian form) by a quantity of the order
$O(t^{t-\nu/2}(|y|+1)^{-(\nu-1)/2})$. Thus the correction to the Gaussian term is comparable with it only in a finite neighborhood of the origin. In the case
$\nu=1$, this correction has the form
$$
\frac{\mathrm{const}}{\sqrt t}\biggl(\operatorname{sign}y\exp\biggl\{-\frac{\mathrm{const}}t(|x|+|y|)^2\biggr\}+O\biggl(\frac 1{|y|}\biggr)\biggr),
$$
i.e., remains of the same order as the Gaussian term on distances
$|y|\sim\sqrt t$. The proof of these results is obtained by a detailed study of the structure of the resolvent
$(\mathcal{T}-zE)^{-1}$ of the stochastic operator
$\mathcal{T}$ of our model for
$z$ lying in a small neighborhood of the point
$z=1$ (the right boundary of the continuous spectrum of
$\mathcal{T}$).
Keywords:
symmetric homogeneous walk on a lattice, Gaussian distribution, stochastic operator and its resolvent, Fredholm formulas, Sokhotzkii formulas. Received: 06.02.1991
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