Take two triangular arrays $X_{N,i}$ and $Y_{N,i}$ of random variables where $1 \le i \le N$. Suppose that the families $\{X_{N,i}\}$ and $\{Y_{N,i}\}$ are independent, and that the following conditions hold:

• Each row is permutation invariant, that is for $N$ fixed and any $\sigma \in S_N$ $$ (X_{N,1},\ldots,X_{N,N}) \stackrel{d}{=} (X_{N,\sigma (1)},\ldots,X_{N,\sigma(N)}) $$ $$ (Y_{N,1},\ldots,Y_{N,N}) \stackrel{d}{=} (Y_{N,\sigma (1)},\ldots,Y_{N,\sigma(N)}) $$
• There are finite constants $a_n,b_n$ such that the sample moments converge in probability $$ \frac1N \sum_{i=1}^N X_{N,i}^n \xrightarrow{\mathcal{P}} a_n $$ $$ \frac1N \sum_{i=1}^N Y_{N,i}^n \xrightarrow{\mathcal{P}} b_n $$

Then is it true that the sample covariance converges in probability to $a_1b_1$? $$ \frac1N \sum_{i=1}^N X_{N,i}Y_{N,i} \xrightarrow{\mathcal{P}} a_1b_1 $$ I make no assumptions about the dependence within the two families.

I am trying to formulate the problem in $L^0(\Omega)$, over the space of measurable function with topology given by convergence in probability. The equivalent form in $L^\omega(\Omega) = \bigcap_{p\ge 1} L^p(\Omega)$, where convergence is with respect to all $\mathbb{E}[|\cdot|^n]$, is easy since $$ \mathbb{E}\left[ \frac1N \sum_{i=1}^N X_{N,i}Y_{N,i} \right] = \frac1N \sum_{i=1}^N \mathbb{E}\left[ \frac1N\sum_{j=1}^N X_{N,j} \right] \mathbb{E}\left[ \frac1N\sum_{k=1}^N Y_{N,k} \right] \to a_1b_1 $$