Let $(A_k)_{k \in I}, (B_k)_{k \in I}, (C_k)_{k \in I}, (D_k)_{k \in I}$ be sequences of random variables with $I$ being a finite index set. Assume we know the following concentration results:
$\mathbb{P}(|A_k - C_k| \geq \varepsilon) \leq K_k^{(1)}$ for all $k$
$\mathbb{P}(|B_k - D_k| \geq \varepsilon) \leq K_k^{(2)}$ for all $k$
where $K^{(1)}, K^{(2)}$ are constants depending on $k$ and $\varepsilon > 0$.
I would now like to get an uppper bound on $$\mathbb{P}\left(\left|\sum_{k\in I} A_k B_k - C_k D_k \right| \geq \varepsilon\right) \leq ??$$
Is there a way to use the two concentration results I have to get a decent bound? If we were not considering the measure, I don't see how I could split up the sum cleverly to to get the differences I know something about, but perhaps there is some property of the measure I could use that I have so far missed? Also, I feel like the finiteness of the index set and thus the sum is important, but I don't really know how to proceed.