Kearns' statistical query model is a well-known learning model with noise tolerance. The statistical query oracle takes as input a statistical query of the form $\{\chi, \tau\}$. Here $\chi$ is any mapping of a labeled example to $\{0, 1\}$ and $\tau \in [0, 1]$ is called the noise tolerance. The oracle returns an estimate for the expectation $\mathbf{E}\chi$, that is, the probability that $\chi = 1$. The additive error of this estimate is at most $\tau$.
The statistical query oracle only guarantees an upper bound of the noise and Kearns' model mentions nothing about the noise distribution. Is there any query oracle in computational learning theory that can answer a statistical query with normal distributed noise? That is, the oracle takes as input a statistical query of the form $\{\chi, \sigma^2\}$ and returns the estimate of $\mathbf{E}\chi$ with additive error of Gaussian distribution $\mathcal{N}(0, \sigma^2)$.
I think this model is reasonable because Gaussian noise is a common assumption in many problems. Also, for central limit theorem, the estimate will converge to a Gaussian distribution (but we are not able to control the variance).