Question Setup

Suppose $p_n(x,y)$ is a sequence of probability densities on $\mathbb R^2$ and $q_n(x)$ is a sequence of densities on $\mathbb R$ such that \begin{align*} \int b(x,y) \ p_n(x,y) \ dx \ dy \to \int b(x,y) p_0(x,y) \ dx \ dy, \end{align*} and \begin{align*} \int c(x) q_n(x) \ dx \to \int c(x) q_0(x) \ dx, \end{align*} for every pair of bounded and continuous functions $b(x,y)$ and $c(x)$. In the language of probability theory, we are saying that if $(X_n, Y_n) \sim p_n$ and $(X, Y) \sim p_0$, then $(X_n, Y_n) \leadsto (X,Y)$ where "$\leadsto$" denotes convergence in distribution. Similarly if $X_n \sim q_n$ and $X \sim q_0$ then $X_n \leadsto X$.

Now, let $r_n(x)$ and $r_0(x)$ be the marginal densities of $\int p_n(x,y) \ dy$ and $\int p_0(x,y) \ dy$. Use these to define densities \begin{align*} s_n(x,y) = \frac{q_n(x)}{r_n(x)} \times p_n(x,y), \qquad s_0(x,y) = \frac{q_0(x)}{r_0(x)} \times p_0(x,y). \end{align*}

My Question

My aim is to show that if $(X_n, Y_n) \sim s_n(x,y)$ and $(X, Y) \sim s_0(x,y)$, then $(X_n, Y_n) \leadsto (X, Y)$. I know that, in general, this won't happen. For example, I know this won't happen if $r_0(x)$ has a smaller support than $q_0(x)$; similarly, I should expect that if $r_0(x)$ has a tails which decay faster than $q_0(x)$ this won't happen. Therefore, I am willing to add the following assumption: \begin{align*} \sup_x \frac{q_0(x)}{r_0(x)} \le M, \qquad \text{for some $M \ge 0$}. \end{align*} Equivalently, I am asking that \begin{align*} \int b(x,y) \frac{q_n(x)}{r_n(x)} p_n(x,y) \ dx \ dy \to \int b(x,y) \frac{q_0(x)}{r_0(x)} p_o(x,y) \ dx \ dy, \end{align*} for all bounded and continuous functions $b(x,y)$.

If what I have given is not sufficient to prove the desired result, I am also looking for conditions under which this does hold - feel free to add any necessary conditions on $p_0(x,y)$ and $q_0(x)$ that you like. If my aim is hopeless, this is of interest as well.

It is essential that there be no assumptions on $p_n$ and $q_n$ beyond the ones I've given. For example, it is unacceptable to assume that there exists a constant which bounds the ratios $q_n(x) / r_n(x)$ (of course, the result is then trivial by dominated convergence). Another example of an unacceptable condition would be that the convergence occurs pointwise in the densities. Only restrictions on $p_0$ and $q_0$ are permissible. Bonus points if you can weaken my current assumption.


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