论文标题
浆果 - 载有矢量价值的马丁纳斯
A Berry-Esseen Bound for Vector-valued Martingales
论文作者
论文摘要
本注释提供了条件性贝里 - 埃森（Berry-Esseen），以限制为martingale差异序列$ \ {x_i \} _ {i = 1}^n $ in $ \ mathbb {r}^d $，$ d \ ge 1 $，适用于过滤;我们近似$ s = \ sum_ {i = 1}^n x_i $的条件分布给定一些$σ$ - field $ \ mathcal {f} _0 \ subset \ subset \ subset \ mathcal {f} _1 $，均为均等的正常随机向量，具有相同的条件方差，给定$ \ sathcal $ s $ s $ s $ s $ s $ s $ s $ s $ s $ s $ s $ s $ s。假设条件差异$ \ MATHSF {e} [x_ix_i^{\ top} \ mid \ mathcal {f} _ {i-1}] $，$ i \ ge 1 $，是$ \ Mathcal {f} 1 $，给定的$ \ Mathcal {f} _0 $是统一的，我们在条件的kolmogorov距离$ s $及其近似情况下给定$ \ Mathcal {f} _0 $，这是$ o_ {a.s。}（a.s。}（a.s.}（a.s.}（ed \ ln（ed \ ln（ed）），
This note provides a conditional Berry-Esseen bound for the sum of a martingale difference sequence $\{X_i\}_{i=1}^n$ in $\mathbb{R}^d$, $d\ge 1$, adapted to a filtration $\{\mathcal{F}_i\}_{i=1}^n$. We approximate the conditional distribution of $S=\sum_{i=1}^n X_i$ given some $σ$-field $\mathcal{F}_0\subset \mathcal{F}_1$ by that of a mean-zero normal random vector having the same conditional variance given $\mathcal{F}_0 $ as the vector $S$. Assuming that the conditional variances $\mathsf{E}[X_iX_i^{\top}\mid\mathcal{F}_{i-1}]$, $i\ge 1$, are $\mathcal{F}_0$-measurable and non-singular, and the third conditional moments of $\|X_i\|$, $ i\ge 1 $, given $\mathcal{F}_0$ are uniformly bounded, we present a simple bound on the conditional Kolmogorov distance between $S$ and its approximation given $\mathcal{F}_0$ which is of order $O_{a.s.}([\ln(ed)]^{5/4}n^{-1/4})$.
