Let $X$ be $n$-dimensional s.t. $X_j\sim Poiss(\lambda_j)$. The components are independent, but the rates are different.

I am interested in bounds for $\Pr(||X-\lambda||\geq y)$, where $\lambda$ is the vector of rates and $||\cdot||$ is some norm. I can use the $1$-norm, but other norms are welcome too.

I bet there are many results out there, but I couldn't find the right term. Can somebody provide pointers?

In particular, I don't want to derive the Chernoff bounds myself, since I'm sure people spent a lot of time trying to come up with good bounds.



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