Large Deviation Principle for multivariat Gaussian

Question

Let us consider the law $\mu_{\epsilon}$ of a multivariat normally distributed random vector X~$\mathcal{N}(0, \epsilon I)$ with mean vector [0,...,0] and covariance Matrix equal to the identity matrix times $\epsilon>0$. Does the Family $\left\{\mu_{\epsilon}:\epsilon>0\right\}$ satisfy a Large Deviation Principle? I know it should be the case for the univariate normal Distribution where one can Show this via Cramers Theorem.

Answer 1

Yes. This is an easy consequence of (say) the .Gärtner-Ellis Theorem. In your case the relevant cumulant generating function is easy to work out.