I was hoping somebody could help me confirm if the following is true.
I have a stochastic process that I simulate in $2$ different modes: without optimizing and with optimizing. If the output of each mode has a reward that is iid, I can apply the central limit theorem to the mean reward for each mode.
My question is, if I take the ratio of the optimized rewards and the non-optimized rewards, can I still apply the central limit theorem to this ratio? To further clarify why I am doing this, I have $100$ samples from each population. Sample $1$ from the optimized population was simulated with the same random seed as sample 1 from the non-optimized population, and so forth. Thus, I want to know how good the optimization is relative to the non-optimized system on a sample basis, which is why I take the ratio of the samples from the optimized population and the samples from the non-optimized population.
As an example, say my samples are:
Population $A$ = Non-Optimized System
Population $B$ = Optimized System
Table of rewards:
| Random Seed | Reward in $A$ | Reward in $B$ | Ratio |
|---|---|---|---|
| $1$ | $10$ | $15$ | $1.5$ |
| $2$ | $15$ | $30$ | $2.0$ |
...etc
I know I can apply CLT to the second and third columns, but can I apply it to the last one?