Okay, given a multinomial random vector $$X=\text{Multinom}(n,\;p_{1},\;\dots,\;p_{k}),$$ so that $$X=(X_{1},\;\dots,\;X_{k})\;\;\;\text{with} \;\;\;\sum_{i=1}^{k}X_{i}=n,$$ I'm looking for a bound on $$\textbf{P}(\sum_{i=1}^{k}|X_{i}-np_{i}| \geq \lambda\sqrt{n}).$$
I know that such an inequality exists (in some obscure textbook) but I'm having trouble finding it. (I had the specific document downloaded, but my computer crashed and I can't find it again.) If anybody knows where I could find such a result, I'd really appreciate the help.
Found it. We have
$$\textbf{P}(\sum_{i=1}^{k}|X_{i}-n p_{i}| \geq 2 \lambda \sqrt{n}) \leq 2^{k}\exp(-2\lambda^{2})$$
for any real $\lambda$. Apparently this is known as the Bretagnolle-Huber-Carol inequality. I'm getting it from the Appendix of "Weak Convergence and Empirical Processes" by van der Vaart and Wellner. My original reference was this post from mathoverflow: https://mathoverflow.net/questions/14560/statistical-approach-to-multinomial-distribution