Consider the following importance sampling estimate:
$$\mathbb{E}_{x \sim P}[\ell(x)] = \mathbb{E}_{x \sim Q}\left[\frac{P(x)}{Q(x)}\ell(x)\right] \approx \frac{1}{L}\sum_{k=1}^L\frac{P(x_k)}{Q(x_k)}\ell(x_k)$$
Where $P(x) = \frac{e^{-x}}{Z_p}$ is a discrete distribution, $Z_p$ is the normalization constant, $x_1,...,x_k \ i.i.d{\sim} Q$, and $\ell$ is bounded function, $0\le\ell\le C$.
Given that: $$E_{x\sim Q}\left[\frac{P}{Q}\right] = 1$$
We deduce:
$$\frac{1}{L}\sum_{k=1}^L\frac{P(x_k)}{Q(x_k)} \approx 1$$
This implies:
$$Z_p \approx {Z}^\prime_p = \frac{1}{L}\sum_{k=1}^L\frac{e^{-x_{k}}}{Q(x_k)}$$
I'm interested in understanding the gap or discrepancy between:
$$\frac{1}{L}\sum_{k=1}^L\frac{e^{-x_{k}}/{Z}^\prime_p}{Q(x_k)} \bullet \ell(x_k) \quad \text{and} \quad \mathbb{E}_{x \sim P}[\ell(x)]$$
Is there any known result, possibly akin to the Hoeffding inequality, that characterizes this difference? Any insights or references would be highly appreciated.