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
$$
\mathbb{E}\left[
\frac1N \sum_{i=1}^N X_{N,i}Y_{N,i}
\right]
=
\frac1N \sum_{i=1}^N \mathbb{E}\left[ \frac1N\sum_{j=1}^N X_{N,j} \right]
\mathbb{E}\left[ \frac1N\sum_{k=1}^N Y_{N,k} \right]
\to a_1b_1
$$