The Allan variance can be defined as: $$\sigma^2_y(\tau)=\frac{1}{2} \left\langle (\Delta y)^2 \right\rangle \quad y(t)=\frac{f(t)-f_n}{f_n}$$ with $f(t)$ being the frequency of some clock at time $t$, and $f_n$ being the nominal frequency of that clock.
If $f(t_0)$ is known, is it correct to derive the distribution of $f(t_0+\tau)$ from that relation? $$\sigma^2_y(\tau) \approx \frac{1}{2} (y(t_0+\tau)-y(t_0))^2$$ $$y(t_0+\tau) \sim \mathcal{N}\left(y(t_0),\sqrt{2}\sigma_y(\tau)\right) $$ $$ f(t_0+\tau) \sim \mathcal{N}\left(f(t_0),f_n\sqrt{2}\sigma_y(\tau)\right)$$
