Distance between the noise and the corrupted signal

Question

How can one formalize the fact that the law of $X+Z$ where $X \in \mathbb{R}^d$ is any vector-valued random variable and $Z\sim \mathcal{N}(0, \sigma^2 \mathbf{I}_d)$ closely resembles the law of $Z$ if $\sigma^2$ is sufficiently large ? $X$ and $Z$ are supposed independent. Ideally, I would like to prove that some distance/divergence between the law of $X+Z$ and $Z$ approaches zero as $\sigma^2\to \infty$.

Answer 1

If were Gaussian, for example, with a non-zero mean, $_{+}=_+_≠_.$

$\displaystyle \lim_{\sigma\rightarrow \infty} (\mu_{X+Z} - \mu_{Z})=\mu_X \ne 0.$

So the difference between the law of $X+Z$ and the law of $Z$ does not approach zero.