$\DeclareMathOperator\Bin{Bin}\DeclareMathOperator\Pr{Pr}$I have a series of distributions $D_k=\Bin(3k,\frac{1+k^{-1/3}}{3})$, and a distribution $D_{k,\ell} = k +\Bin(k,\ell)$ parametrized by $\ell\in[0,2k^{-1/3}]$. I want to find a series of distributions $F_k$ over the support of $[0,2k^{-1/3}]$ for which $\lim_{k\rightarrow\infty}\Delta(D_k,E_{\ell \sim F_k}[D_{k,\ell}]) =0$.
By $\Delta$ I mean the total variation distance of the two distributions (i.e., for two distributions $D_1,D_2$ over finite support $S$, $\Delta(D_1,D_2) = \frac{1}{2} \sum_{i \in S } |\Pr_{X\sim D_1}[X=i]-Pr_{X\sim D_2}[X=i]|$).
By $E[D_{k,\ell}]$ I mean the distribution where you first draw $\ell$ according to $F_k$, and then draw according to $D_{k,\ell}$.
An alternative way to think of it, is that I want to create (or converge to) the original distribution using a convex combination of different distributions with smaller variance, but with different means.
Intuitively, it should work since $D_k$ converges to $N(k+k^{2/3},2k/3)$ so taking $F_k$ to be something like $N(k^{-1/3},2/3k)$ and cut it in the range $[0,2k^{-1/3}]$ should work, but I am not sure how to formalize this argument.
