Normal approximation for the polynomial functionals of correlated random field sampling along random walk path in dimension $1+1$

Abstract

Let $\xi$ be the stationary occupation field generated by a Poisson system of independent simple symmetric random walks on $\mathbb Z$ in space--time dimension $1+1$. For a finite set $A\subset\mathbb Z$, we consider the classical fixed-region observables $W_N(A)$, the cumulative occupation of $A$ up to time $N$, and $D_N(A)$, the number of distinct particles visiting $A$ up to time $N$. We prove quantitative central limit theorems for both observables, with Wasserstein rate of order $N^{-1/4}$. In addition, we introduce an independent nearest-neighbour random walk $S=(S_n,\,n\ge 0)$ on $\mathbb Z$ with non-zero drift and sample the field along this ballistic path. For a fixed polynomial observable $\varphi(x)=\sum_{j=0}^k \beta_j x^j, \beta_k\neq 0$, of degree $k\in \mathbb N$, we consider the partial sums $Y_{N,\varphi}=\sum_{n=1}^N \varphi(\xi(n,S_n)).$ We prove a Wasserstein bound of order $N^{-1/2}$ for the normal approximation of the standardized $Y_{N,\varphi}$. To the best of our knowledge, this is the first quantitative normal approximation result for polynomial functionals of the Poisson occupation field sampled along a random walk path. The drift induces an effective decorrelation of the sampled environment, leading to a substantial improvement over fixed-region sampling. The proofs rely on a representation of $\xi$ as a Poisson functional on path space and on the Malliavin--Stein method for Poisson functionals.