It is well known that for OLS estimators, the parameters are asymptotically normal, i.e. for the regression $y_i = \beta_i x_i$,
$$\hat{\beta_i} \sim \mathcal{N}(\beta_i, \sigma^2 (X_i^T X_i)^{-1})$$
Consequently, for any fixed $x_i$, the product $\hat{\beta_i} x_i$ is also normally distributed.
Is there any analogous relationship for the parameters of a logistic regression?
For context, I am looking to give a bound on the probability that the resultant output from a LR is within a certain interval.