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, only that they are independent from one another.


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The motivation for the problem comes from the following observation. If the random variables have finite moments and if there were finite constants $a',b'$ such that
$$
\mathbb{E}\left[ \frac1N \sum_{i=1}^N X_{N,i} \right] \to  a'
$$
$$
\mathbb{E}\left[ \frac1N \sum_{i=1}^N Y_{N,i} \right] \to  b'
$$
then by the permutation invariance of the families it follows that
$$
\mathbb{E}\left[ \frac1N \sum_{i=1}^N X_{N,i}Y_{N,i}  \right]
=
\frac1N \sum_{i=1}^N \mathbb{E}[X_{N,i}]\mathbb{E}[Y_{N,i}]
$$
$$
=
\frac1N \sum_{i=1}^N \mathbb{E}\left[\frac1N \sum_{j=1}^N X_{N,j}\right]\mathbb{E}\left[ \frac1N \sum_{k=1}^N Y_{N,k}\right] 
\to a'b'
$$
I am asking if a similar phenomenon holds in the case where all quantities are described by convergence in probability rather than convergence of expectations.