I am confused about the following problem:
Given that $(p^n)_n$ is a sequence of densities in $L^2$ which converges weakly to $p^0$, and
$\delta_n(x, x') := \sqrt{\frac{n}{2\pi}}\exp\left( - \frac{n (x-x')^2}{2} \right)$.
The question is, whether $\int \delta_n(x,x')p^n(x')dx'$ converges to $p^0$ in some sense?
I know $\delta_n$ converges to the dirac delta function $\delta$ in the sense of distribtutions, and the above statement seems right, but I don't know how to prove it. Can someone help me? Thanks a lot.
Here are some thoughts about the question which would be too long for a comment.
Denote $h_n(x):=\int \delta(x,x')p^n(x')dx=g_n\star p^n$ with $g_n(x)=\sqrt n(2\pi)^{-1/2}\exp(-x^2/2)$.
Firstly, using Young's inequality for convolution with $r=q=2$ and $p=1$, and boundedness in $\mathbb L^2$ of $(p^n)_n$ we obtain that the sequence $(h_n)_{n\geqslant 1}$ is bounded in $\mathbb L^2$, hence it admits a weakly convergent subsequence.
Second, the result is true if $\lVert p^n-p\rVert_2\to 0$ using again Young's inequality and Fourier transform.