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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | https://arxiv.org/abs/2511.02033 |
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| _version_ | 1866918256612212736 |
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| author | Zaitsev, Andrei Yu. |
| author_facet | Zaitsev, Andrei Yu. |
| contents | Let $X_1,\ldots,X_n$ be $d$-dimensional independent random vectors bounded with probability one. For simplicity, we assume that they have zero mean values: \begin{equation} \mathbf{P}\{\|X_{j}\|\leτ\}=1,\quad\mathbf{E}\,X_{j}=0,\quad j=1,\ldots, n.\nonumber \end{equation} We study the distribution behavior of the sum $S=X_{1}+\cdots+X_{n}$ as a function of the bounding value $τ$.
From the non-uniform Bikelis estimate in the one-dimensional central limit theorem it follows that $$ W_1(F,Φ_σ)\le cτ. $$ with an absolute constant $c$, where $W_1$ is the Kantorovich--Rubinstein--Wasserstein transport distance, $F$ is the distribution of the sum $S$, and $Φ_σ$ is the corresponding normal distribution. The main result of the paper is significantly stronger and more precise. It is claimed that $$ ρ(F,Φ_σ) =\inf\int\exp(|x-y|/cτ)\,dπ(x,y)\le c, $$ where the infimum is taken over all bivariate probability distributions $π$ with marginal distributions $F$ and $Φ_σ$. The result has also been generalized to distributions with sufficiently slowly growing cumulants from the class $\mathcal{A}_{1}(τ)$, introduced in the author's 1986 paper. The possibility of generalizing the result to the multivariate case is discussed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_02033 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Estimates of transport distance in the central limit theorem Zaitsev, Andrei Yu. Probability 60F Let $X_1,\ldots,X_n$ be $d$-dimensional independent random vectors bounded with probability one. For simplicity, we assume that they have zero mean values: \begin{equation} \mathbf{P}\{\|X_{j}\|\leτ\}=1,\quad\mathbf{E}\,X_{j}=0,\quad j=1,\ldots, n.\nonumber \end{equation} We study the distribution behavior of the sum $S=X_{1}+\cdots+X_{n}$ as a function of the bounding value $τ$. From the non-uniform Bikelis estimate in the one-dimensional central limit theorem it follows that $$ W_1(F,Φ_σ)\le cτ. $$ with an absolute constant $c$, where $W_1$ is the Kantorovich--Rubinstein--Wasserstein transport distance, $F$ is the distribution of the sum $S$, and $Φ_σ$ is the corresponding normal distribution. The main result of the paper is significantly stronger and more precise. It is claimed that $$ ρ(F,Φ_σ) =\inf\int\exp(|x-y|/cτ)\,dπ(x,y)\le c, $$ where the infimum is taken over all bivariate probability distributions $π$ with marginal distributions $F$ and $Φ_σ$. The result has also been generalized to distributions with sufficiently slowly growing cumulants from the class $\mathcal{A}_{1}(τ)$, introduced in the author's 1986 paper. The possibility of generalizing the result to the multivariate case is discussed. |
| title | Estimates of transport distance in the central limit theorem |
| topic | Probability 60F |
| url | https://arxiv.org/abs/2511.02033 |