# Concentration/Tail Bounds for a vector of Poisson r.v.

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.

Thanks!