Suppose I have a ratio \begin{equation} R=\frac{S_X+S_Y}{S_Z} \end{equation} Where $S_X$ is a sub-Gaussian variable, such that $S_X=X_1+X_2+...+X_n$ with $a_i\leq X_i\leq b_i$ almost surely, and $S_Z$ is of quadratic form defined in a similar manner. To obtain an upper bound for $R$, I can use Hoeffding's inequality for $S_X$ - i.e., \begin{equation} P\left[S_X\leq\mathbb{E}[S_X]-t_1 \right]\leq \exp\left(-\frac{2t^2}{\sum_{i=1}^n(b_i-a_i)^2}\right). \end{equation}
and, for instance, for the denominator I could use the Hanson-wright inequality to lower bound the denominator, and end up with an expression of the form: \begin{equation} R\leq \frac{\mathbb{E}[S_X]-t_1+Y}{\mathbb{E}[S_Z]+t_2} \end{equation} with some probability. However, that probability would be the combination of the Hoeffding's inequality and Hanson-Wright inequalities used to replace two of the variables with their respective upper and lower bounds. How can I combine the probabilities when we have a ratio? Can Union bounds be used in such circumstances? (e.g., Boole's inequality)