To give some context, suppose you have a sequence of random variables $X_n$ that you know weakly converges to a random variable $X$. Suppose you also have a sequence of random variables $Y_n$. Then, if $d(.,.)$ denotes the Levy metric between two probability distributions, we have that $ d(F_{Y_n},F_X)\leq d(F_{X_n},F_{Y_n})+d(F_{X_n},F_X)$. We know that $d(F_{X_n},F_X)\to 0$ as $n\to\infty$. If we can also have that $d(F_{X_n},F_{Y_n})\to0$ then we can conclude $Y_n\to X$ weakly.

I am trying to understand, given $Y_n$, how difficult it is to "construct" a sequence of random variables $X_n$ such that $X_n\to X$ weakly and $X_n,Y_n$ are close to each other. I haven't seen many examples in this regard.

A classic example would be the delta method used in statistics: if $f$ is a $C^1$ function and $X_1,...,X_n$ are iid with mean $\mu$ and variance $\sigma^2$ and with the assumption that $f'(\mu)\neq 0$ then $\dfrac{\sqrt{n}(f(\overline{X_n})-f(\mu))}{f'(\mu)\sigma}\to N(0,1)$ weakly.

What one does in the above case is that one does a Taylor series decomposition to have $\dfrac{\sqrt{n}(f(\overline{X_n})-f(\mu))}{f'(\mu)\sigma}=\dfrac{\sqrt{n}(\overline{X_n}-\mu)}{\sigma}+\sqrt{n}R_n$. Then we show $\sqrt{n}R_n\to 0$ in probability, and thus we are approximating an apriori complicated function $\dfrac{\sqrt{n}(f(\overline{X_n})-f(\mu))}{f'(\mu)\sigma}$ by the relatively simple object $\dfrac{\sqrt{n}(\overline{X_n}-\mu)}{\sigma}$ which by an application of CLT, will go to $N(0,1)$ weakly.

So given $Y_n$ (in our case $\dfrac{\sqrt{n}(f(\overline{X_n})-f(\mu))}{f'(\mu)\sigma}$) we produced $X_n$ (in our case $\dfrac{\sqrt{n}(\overline{X_n}-\mu)}{\sigma}$) and we showed the convergence in law.

I am interested in an even wider context. Consider a Wigner matrix $X_n$ then it is classical that $ESD(X_n/\sqrt{n})\to \rho(.)$ where $\rho(.)$ is standard semi circle law. We also know that if $X$ is a $p\times n$ iid matrix where $p,n\to\infty$ with $p/n\to0$, then $ESD(\sqrt{\dfrac{n}{p}}(\dfrac{XX'}{n}-I_p))\to \rho(.)$ as well.

I was wondering if one can try to see if one can find a Wigner matrix $Z_n$ constructed from $X$ such that we can conclude convergence of $ESD(\sqrt{\dfrac{n}{p}}(\dfrac{XX'}{n}-I_p))$ by approximation by $ESD(Z_n/\sqrt{n})$. This will help in finding more universality results and explore the structure of a random matrix as a whole.

NOTE: In the context of random matrices, the Mallow's metric is mostly used. I know of some inequalities that help to show that it does not matter if you do not center the sample covariance matrix (can't recall the name) but not really any result in the direction I am looking for.