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In theCommunicationProbabilistic channel modeling previous section, we explored an important theorem that underpins the Gaussian modelingGaussian modeling of numerous random quantities of interest: the Central Limit Theorem (CLT)Central Limit Theorem (CLT). The CLT pertains to an i.i.d. random processI.i.d. random process  \(\{X_i \}_{i=1}^{n}\) . For the sake of illustration, we considered a simple yet generalizable setup in which \(\mathbb {E}[X_i] = 0\) and \(\textsf{var}(X_i) = \frac{1}{n}\) . In this context, the CLT can be stated as: \( Z_n := X_1 + X_2 + \cdots + X_n \;\overset{\text {in dist}}{\longrightarrow } \; Z \sim \mathcal{N} (0,1)\) where the notation \(\overset{\text {in dist}}{\longrightarrow }\) signifies the convergence in distributionConvergence in distribution: To prove the CLT, we relied on two key claims.

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Information Technology Applications

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  • 发表日期：

    2025年

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  • Changho Suh

摘要

In theCommunicationProbabilistic channel modeling previous section, we explored an important theorem that underpins the Gaussian modelingGaussian modeling of numerous random quantities of interest: the Central Limit Theorem (CLT)Central Limit Theorem (CLT). The CLT pertains to an i.i.d. random processI.i.d. random process  \(\{X_i \}_{i=1}^{n}\) . For the sake of illustration, we considered a simple yet generalizable setup in which \(\mathbb {E}[X_i] = 0\) and \(\textsf{var}(X_i) = \frac{1}{n}\) . In this context, the CLT can be stated as: \( Z_n := X_1 + X_2 + \cdots + X_n \;\overset{\text {in dist}}{\longrightarrow } \; Z \sim \mathcal{N} (0,1)\) where the notation \(\overset{\text {in dist}}{\longrightarrow }\) signifies the convergence in distributionConvergence in distribution: To prove the CLT, we relied on two key claims.