Take two triangular arrays $x_{N,i}$ and $y_{N,i}$ for $1 \le i \le N$. Suppose that the $\{x_{N,i}\}$ and $\{y_{N,i}\}$ are independent, and that the sample means both converge in probability to zero $$ \frac1N \sum_{i=1}^N x_{N,i} \xrightarrow{\mathcal{P}} 0 $$ $$ \frac1N \sum_{i=1}^N y_{N,i} \xrightarrow{\mathcal{P}} 0 $$ Suppose also that sample moments of every order $n \ge 2$ converge in probability to some finite constants $$ \frac1N \sum_{i=1}^N |x_{N,i}|^n \xrightarrow{\mathcal{P}} a_n < \infty $$ $$ \frac1N \sum_{i=1}^N |y_{N,i}|^n \xrightarrow{\mathcal{P}} b_n < \infty $$ Then is it true that the sample covariance converges in probability to zero? $$ \frac1N \sum_{i=1}^N x_{N,i}y_{N,i} \xrightarrow{\mathcal{P}} 0 $$ In the case where the moments of the $x_{N,i}$ and $y_{N,i}$ exist it is easy to show, but in general I am struggling to find a reference.