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[LLN CLT] Update Translations #49

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nisha617 merged 3 commits into from
Aug 17, 2025
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[LLN CLT] Update Translations #49

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4 changes: 2 additions & 2 deletions lectures/lln_clt.md
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Expand Up @@ -717,14 +717,14 @@ plt.show()

这种标准化可以基于以下三个观察结果来实现。

首先，如果$\mathbf X$是$\mathbb R^k$中的随机向量，且$\mathbf A$是常数且为$k \times k$矩阵，那么
首先，如果$\mathbf X$是$\mathbb R^k$中的随机向量，$\mathbf A$是常数且为$k \times k$矩阵，那么

$$
\mathop{\mathrm{Var}}[\mathbf A \mathbf X]
= \mathbf A \mathop{\mathrm{Var}}[\mathbf X] \mathbf A'
$$

其次，根据[连续映射定理](https://en.wikipedia.org/wiki/Continuous_mapping_theorem)，如果$\mathbf Z_n \stackrel{d}{\to} \mathbf Z$在$\mathbb R^k$中成立，且$\mathbf A$是常数且为$k \times k$矩阵，那么
其次，连续映射定理指出， 如果$g(\cdot)$是一个连续函数， 且随机变量序列$\{\mathbf{Z}_n\}$依分布收敛到随机变量$\mathbf{Z}$, 那么$\{g(\mathbf{Z}_n)\}$也依分布收敛到随机变量$g(\mathbf{Z})$。根据连续映射定理，如果$\mathbf Z_n \stackrel{d}{\to} \mathbf Z$在$\mathbb R^k$中成立，且$\mathbf A$是常数且为$k \times k$矩阵，那么
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Copilot AI Aug 16, 2025

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There are inconsistent spacing issues around punctuation in the Chinese text. There should be no space before commas (， should be ，) and the spacing around mathematical notation should be consistent.

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其次，连续映射定理指出， 如果$g(\cdot)$是一个连续函数， 且随机变量序列$\{\mathbf{Z}_n\}$依分布收敛到随机变量$\mathbf{Z}$, 那么$\{g(\mathbf{Z}_n)\}$也依分布收敛到随机变量$g(\mathbf{Z})$。根据连续映射定理，如果$\mathbf Z_n \stackrel{d}{\to} \mathbf Z$在$\mathbb R^k$中成立，且$\mathbf A$是常数且为$k \times k$矩阵，那么
其次，连续映射定理指出，如果$g(\cdot)$是一个连续函数，且随机变量序列$\{\mathbf{Z}_n\}$依分布收敛到随机变量$\mathbf{Z}$, 那么$\{g(\mathbf{Z}_n)\}$也依分布收敛到随机变量$g(\mathbf{Z})$。根据连续映射定理，如果$\mathbf Z_n \stackrel{d}{\to} \mathbf Z$在$\mathbb R^k$中成立，且$\mathbf A$是常数且为$k \times k$矩阵，那么

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@mmcky mmcky Aug 17, 2025

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@nisha617 if you agree with this -- you can select Commit suggestion, but feel free to ignore it if it isn't of value -- just click Resolve conversation. Thanks.


$$
\mathbf A \mathbf Z_n
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