Normal approximation for the polynomial functionals of correlated random field sampling along random walk path in dimension $1+1$
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| Format: | Preprint |
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2026
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| _version_ | 1866908889463652352 |
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| author | Huang, Ao Rang, Guanglin Su, Zhonggen |
| author_facet | Huang, Ao Rang, Guanglin Su, Zhonggen |
| contents | Let $ξ$ 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 $φ(x)=\sum_{j=0}^k β_j x^j, β_k\neq 0$, of degree $k\in \mathbb N$, we consider the partial sums $Y_{N,φ}=\sum_{n=1}^N φ(ξ(n,S_n)).$ We prove a Wasserstein bound of order $N^{-1/2}$ for the normal approximation of the standardized $Y_{N,φ}$. 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 $ξ$ as a Poisson functional on path space and on the Malliavin--Stein method for Poisson functionals. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_15308 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Normal approximation for the polynomial functionals of correlated random field sampling along random walk path in dimension $1+1$ Huang, Ao Rang, Guanglin Su, Zhonggen Probability 60F05, 60G55, 60H07, 60J10, 60K35 Let $ξ$ 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 $φ(x)=\sum_{j=0}^k β_j x^j, β_k\neq 0$, of degree $k\in \mathbb N$, we consider the partial sums $Y_{N,φ}=\sum_{n=1}^N φ(ξ(n,S_n)).$ We prove a Wasserstein bound of order $N^{-1/2}$ for the normal approximation of the standardized $Y_{N,φ}$. 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 $ξ$ as a Poisson functional on path space and on the Malliavin--Stein method for Poisson functionals. |
| title | Normal approximation for the polynomial functionals of correlated random field sampling along random walk path in dimension $1+1$ |
| topic | Probability 60F05, 60G55, 60H07, 60J10, 60K35 |
| url | https://arxiv.org/abs/2603.15308 |