I have the following probability bound I want to prove but don't know what bound the author is using.
say $N_k \sim \mathcal{N}(0,\sigma^2)$
Then, we are interested in upper bounding the probability:
$\mathbb{P}\left( \sum_{k=1}^n N_k^2 \geq n^3 \right)$.
The resulting upper bound in the paper is :
$\mathbb{P}\left( \sum_{k=1}^n N_k^2 \geq n^3 \right) \leq \frac{\sigma^2}{n^2}$
Any clue or material you guys may refer me to?
What I tried, no too fancy, just dividing both sides by $\sigma^2$, then we obtain a sum of squared normal Gaussian variables, and thus, a chi-squared. Then:
$\mathbb{P}\left( \sum_{k=1}^n N_k^2 \geq n^3 \right) = \mathbb{P}\left( {\chi}^2_n \geq \frac{n^3}{\sigma^2} \right)$.
Then, don't know how to proceed ??
Thanks