Sanov's theorem and Dvoretzky–Kiefer–Wolfowitz's inequality tell us how fast the empirical distribution concentrates around the true underlying probabilty distribution.

What is known about the concentration of the posterior distribution ? Do such inequalities exist for a mode, a median, the mean or a sample of the posterior ?

Formally, let $\left\lbrace \mathbb{P}_{\theta} \ : \ \theta \in \Theta \right\rbrace$ be a set of probability distributions on $\mathbb{R}$ and $\mu_0$ a prior on $\Theta$.

Let $\mu_n$ be the posterior distribution, i.e. the regular conditional distribution of $\theta$ given $X_1,...,X_n$ when the distribution of $(\theta,X_1,...,X_n)$ is $\mathbb{P}=\mu_0 \otimes \mathbb{P}_{\theta}^{\otimes n}$. $$\mu_n(A,x_1,...,x_n) = \mathbb{P}(\theta \in A \ | \ X_1 = x_1,...,X_n = x_n)$$ If the model is true, this is indeed the distribution of $\theta$ when one has observed n i.i.d samples $(X_1,...,X_n)$ with values $(x_1,...,x_n)$. Even if the model is false, given $n$ values $(x_1,...,x_n)$, $A \mapsto \mu_n(A,x_1,...,x_n)$ still defines a probability distribution on $\Theta$.

Now if I'm a frequentist, I do not agree that $\theta$ is a random variable, I believe it is a parameter with a true (but unknown) value $\theta_{\text{true}}$ and the data $(X_1,...X_n)$ are then i.i.d with true underlying (but unknown) distribution $\mathbb{P}_{\theta_{\text{true}}}^{\otimes n}$. I then a get a random probability distribution $A \mapsto \mu_n(A,X_1,...,X_n)$ on $\Theta$ for which I can take a mode, a median (if $\Theta$ is a subset of $\mathbb{R})$), the mean (also if $\Theta$ is a subset of $\mathbb{R})$) or even a sample (which are all random quantities because they depend on the data).

I'm looking for finite-time upper bounds on the quantities $\mathbb{P}_{\theta_{\text{true}}}^{\otimes n}\left( \bullet \notin A \right)$ where $\bullet$ stands for the mean, a median, a mode or a sample of the posterior $\mu_n$ and $A$ is a measurable subset of $\Theta$ containing $\theta_{\text{true}}$.

I don't know much about bayesian statistics and would be happy with results in any setting ($\Theta$ can be as simple as you want, like a standard exponential family, or even the family of Bernoulli distributions, and $A$ can be any specific neighbourhood of $\theta_{\text{true}}$).