I am not very familiar with rigorous knowledge about convergence and asymptotic theory in statistics. Although my question is more statistics-related, my problem arises from the lack of knowledge about math regarding the difference between convergence in probability and $L^2$ convergence.
More precisely, if I have the following convergence: $$||X(\hat{\beta}-\beta )||_2^2 =o_p(1)$$
where $X$ is a $n\times p$ matrix, $\beta,\hat{\beta}$ are $p\times 1$ respectively and $||y||_2^2 $ is defined as $\sum_{i=1}^{n}y_i^2$. Hence, in that case, just a bracket squared (in that context the prediction error squared)
is this the same as saying $$X(\hat{\beta}-\beta)\rightarrow0$$ converges to zero in probability.