Suppose that we have a sequence of i.i.d. random variables $X_1,X_2,...$ on $\mathbb{R}^2$ with probability distribution $\mu$ and define random empirical measures as
$$\mu_n=\frac{1}{n}\sum_{k=1}^n \delta_{X_k}$$
We know these almost surely converge weakly to $\mu $ as $ n \rightarrow \infty$. Now suppose that $Y_1,Y_2,...$ is a sequence of independent standard $\mathbb{R}^2$ multivariate normal random variables independent of the sequence $X_1,X_2,...$ I am interested in the distribution of the measures that are the weak limits of the sequence defined as
$$\nu_n=\frac{1}{n}\sum_{k=1}^n (\delta_{X_k}+Y_k)$$
Would such a distribution even exist? Is the set of all outcomes where the weak limit does not exist a null set? Is this problem even well defined? If anyone has any sources on this, they would be greatly appreciated.