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.