We present a new method for proving the norm concentration inequality of sub-Gaussian variables. Our proof is based on an averaged version of the moment generating function, termed the averaged moment generating function. Our method applies to both vector cases to bound the vector norm and matrix cases to bound the operator norm. Compared with the widely adopted $\varepsilon$-net technique-based proof of the sub-Gaussian norm concentration inequality, our method does not rely on the union bound and promises a tighter concentration bound.

翻译：我们提出了一种证明亚高斯变量范数集中不等式的新方法。该证明基于矩生成函数的平均化版本，称为平均矩生成函数。我们的方法既适用于向量情形以界定向量范数，也适用于矩阵情形以界定算子范数。与广泛采用的基于$\varepsilon$-网技术的亚高斯范数集中不等式证明相比，我们的方法不依赖于并集界，并能提供更紧的集中界。