$a_i \in (0,1)$, $I = 1,\cdots,N$ are $N$ random samples from uniform distribution $U(0,1)$. $a_i$ is in ascending order $a_1 < a_2 < \cdots < a_N$.
$Q(p)$, $p\in(0,1)$ is a differentiable quantile function of random variable $X$.
The Monte Carlo integration gives the mean and variance by $\frac{1}{N}\sum_{i=1}^N Q(a_i)$ and $\frac{1}{N}\sum_{i=1}^N Q(a_i)^2$.
Now suppose there is another set of random samples $b_i \in (0,1)$, $I = 1,\cdots,N$ in ascending order from uniform distribution.
$\bf Question:$ do we have $$ \frac{1}{N}\sum_{i=1}^N Q(a_i)Q(b_i) \rightarrow \text{Var(X)} \,\,? $$ Further, do we have asymptotic normality $ \frac{1}{N}\sum_{i=1}^N Q(a_i)Q(b_i) \rightarrow N(\text{Var(X)},D)? $ How do we find the variance $D$?
I tried plugging in Taylor's Theorem $Q(a_i) = Q(b_i) + (b_i - a_i)Q'(b_i) + \frac{1}{2}(b_i - a_i)^2 Q''(b_i) + o(|(b_i - a_i)^2|)$, but couldn't proceed. I am considering fixing $a_i = \frac{i}{N+1}$ for simplicity.