SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\mu^ {n} \left(\frac {1}{n} \sum_ {i = 1} ^ {n} \| x _ {i} \| ^ {2} - \mathbb {E} _ {x \sim \mu} \| x \| ^ {2} \geq 8 \sigma^ {2} \max \left(\sqrt {\frac {\log (1 / \delta)}{n}}, \frac {\log (1 / \delta)}{n}\right)\right) \leq \delta .
$$
This proves the first part of the Lemma. Specifically, we have the following:
$$
\mu^ {n} \left(\frac {1}{n} \sum_ {i = 1} ^ {n} \| x _ {i} \| ^ {2} - \mathbb {E} _ {x \sim \mu} \| x \| ^ {2} \geq 8 \sigma^ {2} \frac {\log (1 / \delta)}{n}\right) \leq \delta \quad \text {i f} \delta \leq e ^ {- n}
$$
$$
\mu^ {n} \left(\frac {1}{n} \sum_ {i = 1} ^ {n} \| x _ {i} \| ^ {2} - \mathbb {E} _ {x \sim \mu} \| x \| ^ {2} \geq 8 \sigma^ {2} \sqrt {\frac {\log (1 / \delta)}{n}}\right) \leq \delta \quad \mathrm {i f} \delta \geq e ^ {- n}
$$
The second inequality proves the last part of the Lemma.