Imagine that I have a sequence of random variables $\mu_n$. For each realization of $\mu_n$ , I construct a random distributed RV $X_n$ such that $X_n\sim N(\mu_n,E)$. Suppose I am given that $\mu_n \xrightarrow{as}\mu$ where $\mu$ is a scalar. Can i say anything about the asymptotic distribution of $X_n$.
my attempt :
It seems to me that $F_n(x)$ will almost surely converge to $F(x)$ where $F(x)$ is the CDF associated with $N(\mu,E)$. This would also hold for for the characteristic function. Does that imply that i can make some statement about the asymptotic distribution of $X_n$. Does it help if i know that $\mu_n\sim N(M,\frac{2}{n})$
First of all: Since mean and CDF are deterministic objects. It doesn't make sense to talk about "almost sure" convergence.
If $X_n$ are Gaussian random variables with mean $\mu_n$ and variance $\sigma_n^2$ such that $\mu_n$ and $\sigma_n^2$ converge as $n \to \infty$, say to $\mu$ and $\sigma^2$, then $X_n$ converges in distribution to a Gaussian random variable $X$ with mean $\mu$ and variance $\sigma^2$.
In the case which you mentioned at the very end of your question, i.e. $\mu_n=M$ is constant and $\sigma_n^2 = 2/n$, then $\mu=M$ and $\sigma^2=\lim_{n \to \infty} 2/n=0$, which means that the limit $X$ has mean $M$ and variance $0$, i.e. $X$ is constant almost surely.