Assume we are given $n$ i.i.d. samples $x_1, x_2, ..., x_n$ of a random vector $X$ with mean 0 and (true) covariance matrix bounded in norm. Does the standard covariance matrix estimator
$$n^{-1} \sum_{i=1}^n x_i x_i^{\top}$$
converge almost surely to the true covariance matrix? I would like to apply Kolmogorv's strong law of large numbers, but all the literature I found only considers univariate random variables explicitly.
Does it also hold for matrices? Could you then please also provide me literature for this?