let $X_n$ be sequence of random variables with the associated distribution $N(\mu_n,\beta_n+E)$. That is a sequence of normally distributed random variables with changing mean and variance. $\beta_n$ is scalar sequence. $\mu_n$ is a sequence of random variables. It is given that $$\mu_n \xrightarrow{as}M$$ where M is scalar and $\lim_{n\to \infty} \beta_n=0$. Can i say that $X_n \xrightarrow{d}N(M,E)$
My approach : If i look at $F_n$, the CDF of $X_n$, that it self is a random variable. Now $F_n(x)$ will converge almost surely to $F(x) \forall x$. The problem is what can I say about the convergence in distribution. My understanding is that $F_n$ is a compound distribution. Is there any result on the convergence of a compound distribution?
