Let $X = \left(Z_1^2 + \dots + Z_N^2 \right)^{1/2}$, where $Z_1, \dots, Z_N$ are $N$ independent, normally distributed random variables with mean 0 and standard deviation $\sigma$. $X$ is distributed according to a Chi distribution. Consider a sequence of $K$ such variables: $X_1, \dots, X_K$, where $X_k$'s are independent.
What can be said about the following probability? $$\mathbb{P}\left[\left\vert \sum_{k=1}^{K} X_k - K\sqrt{N}\sigma\right\vert \ge t\right]$$
$$\begin{align} &= 1- \mathbb{P}\left(\left\vert \sum_{k=1}^{K} X_k - K\sqrt{N}\sigma\right\vert \le t\right)\\ &= 1 - \mathbb{P}\left( \underbrace{ -t + K\sqrt{N}\sigma }_{=\alpha}\le \sum_{k=1}^{K} X_k \le \underbrace{t + K\sqrt{N}\sigma}_{=\beta} \right) \\ &= 1 - \mathbb{P}\left( \alpha - \mu \le \sum_{k=1}^{K} X_k \le \beta - \mu\right) \\ \end{align}$$ where $\left(\mu, \sigma^2 \right)$ are the mean and the variance of $\sum_{k=1}^{K} X_k $ $$\left(\mu, \sigma^2 \right) = \left(K\cdot\sqrt{2}\frac{\Gamma((N+1)/2)}{\Gamma(N/2)},K^2 \cdot \left(N- 2\frac{\Gamma^2((N+1)/2)}{\Gamma^2(N/2)}\right) \right)$$ and then, it suffices to apply the Selberg's inequality to obtain the upper bound.