Consider the setup:
Let $(X_i | \mu = m, \sigma = s)$ be a continuous random variable with pdf$$f_{X_i | \mu, \sigma}(x | m, s) = f_{X_i | \mu , \sigma}\big( \frac{x-m}{s} | 0,1 \big) \ s^{-1}, x \in \mathbb{R}$$ where $( \mu,\sigma)$ is a random location-scale parameter.
Let then $\mathbf{X} = (X_1, ..., X_n)$ be a random vector, for some integer $n$. The pdf of $(\mathbf{X} | \mu = m, \sigma = s)$ at $\mathbf{x} = (x_1, ..., x_n) \in \mathbb{R}^n$ is $f_{\mathbf{X} | \mu, \sigma}(\mathbf{x} | m,s)$.
We observe the realization $\mathbf{x}^* = (x_1^*, ..., x_n^*)$ of $\mathbf{X}$. Assuming a non-informative (improper) prior $f_{\mu,\sigma}(m,s) \propto s^{-1}, (m,s) \in \mathbb{R} \times \mathbb{R}_+$ for $(\mu,\sigma)$, we have posterior \begin{equation} f_{\mu,\sigma |\mathbf{X}}(m,s |\mathbf{x}^*) \propto f_{\mathbf{X}|\mu,\sigma}(\mathbf{x}^* |m,s) \ s^{-1}, (m,s) \in \mathbb{R} \times \mathbb{R}_+, \end{equation} which by definition will be proper if \begin{equation} \int_{-\infty}^{+\infty} \int_{0}^{+\infty} f_{\mathbf{X} |\mu,\sigma}(\mathbf{x}^* |m,s) \ s^{-1} \ \text{d}s \ \text{d}m < \infty. \end{equation}
My question is thus: Are there known conditions on the likelihood $f_{\mathbf{X} | \mu,\sigma}(\mathbf{x}^* |m, s)$ such that the above integral is finite (and hence the posterior $f_{\mu,\sigma | \mathbf{X}}(m,s |\boldsymbol{x}^*)$ is proper) ?
I think that expressing the likelihood using sklar's theorem shall help, i.e. \begin{equation} f_{\mathbf{X} | \mu,\sigma}(\mathbf{x}^* |m, s) = c\big(F_{X_1 | \mu,\sigma}(x_1^* | m, s), ...,F_{X_n | \mu,\sigma}(x_n^* | m, s) \big) \ \prod_{i = 1}^n f_{X_i | \mu, \sigma}(x_i^* | m, s), \end{equation} where $c( \cdot )$ is the density of the copula of $\mathbf{X}$ and $F_{X_i |\mu,\sigma}( x_i^* | m, s)$ the cdf of $(X_i | \mu = m, \sigma = s)$. But I fail to see how to tackle the problem then.
Intuitively, I would say that a necessary but not sufficient condition is that $n \geq 2$. And if $n \geq 2$, then I suspsect that one needs $X_1, ..., X_n$ not to be comonotonic.
