Inverting Gaussian convolution in weak convergence I have the following question regarding weak convergence of probability measures on $\mathbb{R}$. Suppose that $(\mu_n)_n$ is a sequence of such probability measures and that, for each $k\in\mathbb{N}$, a subsequence of the sequence $(\nu^{(k)}_n)_n$ converges weakly to some probability measure $\nu^{(k)}$. Here,
$$ \nu^{(k)}_n:=\mu_n\ast\mathcal{N}(0,k^{-1})$$
is the convolution of $\mu_n$ with a cenetered normal distribution with variance $k^{-1}$.
My question: Does this imply that a subsequence of the original sequence $(\mu_n)_n$ also converges weakly to some probability measure $\mu$?

It is enough if we have convergence for one fixed value of $k$. By Prohorov's Theorem it is enough to show tightness of $(\mu_n)$. Let $X_n \sim \mu_n$ and $Y \sim N(0,\frac 1 k)$ Then, for any given $\epsilon >0$, there exisst $M$ such that $P|X_n+Y|>M) <\epsilon /2$ for all. $n$. We can also choose $M$ such that $P(|Y|>M) <\epsilon/2$. Now $P|X_n|>2M)\leq  P|X_n+Y|>M) +P(|Y|>M) <\epsilon$ for all $n$.
