Let $x=[x^1,\dots,x^p]^T:=x_1$, $y:=xx^T$, $w:=y-Ey$, and $s:=\sum_1^n x_ix_i^T$. 
Then, by the appropriate laws of large numbers, $s/n\to Ey$ almost surely and hence in probability and in distribution, provided that $Ey$ exists in $\mathbb R^{p\times p}$. Everywhere here, the convergence is for $n\to\infty$.