Let $X=(X_1,\ldots,X_n)$, where $X_i \sim P_{p_i}(0,\frac{1}{\lambda})$ are iid, $P_{p_i}$ is sub gaussian distribution for $i^{th}$ element, and 0 and $1/\lambda$ are mean and variance.

I'm looking for a result on the concentration of $\|X\|_2^2$ something of the form $E \|X\|_2^2 \leq c$ with $P(\|X\|_2^2 \geq c+\epsilon)\leq f(\epsilon)$. What happens when all the distribution are normals?

I have asked a similar question on https://math.stackexchange.com/questions/3798100/concentration-of-l2-norm-of-a-vector-sampled-from-a-distribution

Let $X=(X_1,\ldots,X_n)$, where $X_i \sim P_{p_i}(0,\frac{1}{\lambda})$ are iid, $P_{p_i}$ is sub gaussian distribution for $i^\text{th}$ element, and 0 and $1/\lambda$ are mean and variance.

I'm looking for a result on the concentration of $\|X\|_2^2$ something of the form $E \|X\|_2^2 \leq c$ with $P(\|X\|_2^2 \geq c+\epsilon)\leq f(\epsilon)$. What happens when all the distribution are normals?

I have asked a similar question on https://math.stackexchange.com/questions/3798100/concentration-of-l2-norm-of-a-vector-sampled-from-a-distribution