I read in some lecture notes the following definition of contraction rate:
Definition (Posterior rate of contraction) The posterior distribution $\Pi_n\left(\cdot \mid X^{(n)}\right)$ is said to contract at rate $\epsilon_n \rightarrow 0$ at $\theta_0 \in \Theta$ if $\Pi_n\left(\theta: d\left(\theta, \theta_0\right)>M \epsilon_n \mid X^{(n)}\right) \rightarrow 0$ in $P_{\theta_0}^{(n)}$ probability, for a sufficiently large constant $M$ as $n \rightarrow \infty$.
Q: Is there literature that treats results of the type:
For any $\eta >0$, $$\Pi_n\left(\theta: d\left(\theta, \theta_0\right)> \eta \epsilon_n \mid X^{(n)}\right) \rightarrow 0$$ in $P_{\theta_0}^{(n)}$ probability, as $n \rightarrow \infty$.
This is a slightly stronger type of contraction, but I could not find anything in the literature. Any reference is highly appreciated.
What I am really interested in is: under which conditions, under the posterior:
$$ \epsilon_n^{-1} d\left(\theta, \theta_0\right) \rightarrow 0 $$
