Let $(q_n)$ be sequence of probability density functions of the couple $(x,y)\in \mathbb R^2$, $p_n$ is the marginal density of $q_n$, i.e. $p_n(x):=\int q_n(x,y)dy$.
Another sequence of functions $(pb_n)$ is defined by $pb_n(x):=\int b(y)q_n(x,y)dy$, where $0<l\le b(y)<L$, $\forall y\in \mathbb R$.
Given that $(p_n)$ converges to $p_0$ weakly in $L^2$, what do we know about $(pb_n)$? Is there any chance that $(pb_n)$ also converges? It would be fine if some additional conditions need to be added.
The above question was asked last month, I found it very interesting. Is it possible to change the question to such that: can we have a form of $b(y)$ so that both integral converge to the same value but the presence of $b(y)$ makes the convergence faster?
This question is related to the question in this link: Convergence of marginal distribtution
In the link it was asked about the convergence of the sequence when we introduce a term $b(y)$ to $q_n$. My question can we think of existence of the $b(y)$ such that introducing it will not change the convergent value of $q_n$ but will make the convergence faster. If $b(y) =1$ it will not change convergence but the sequence won;t converge fast either. Hope my question clear.