I am interested in computing the asymptotics of the moments of a probability distribution defined by $p(x \mid \lambda) \propto \exp(-\lambda V(x))$ where $V$ is some smooth multivariate function of interest with a unique global minimum that without loss of generality we can consider to be $0$.
Naively (wrongly?) applying the saddle-point method to this gives $\mathbb{E}[X_{\lambda}^k] \approx 0$ when $\lambda \to \infty$, with no information so as to the speed of convergence (because $x^k = 0$ at $0$). Are there finer results in this specific case?
Thanks in advance.