pr.probability – Multivariate CDF limit of coordinated empirical averages of a multivariate distribution

Let $ mathbb P $ to be the law of a multivariate random variable $ X $ in $ mathbb R ^ p $, with the corresponding multi-variable CDF defined by $ F_X (z ^ {(1)}, ldots, z ^ {(p)}): = mathbb P (X ^ {(1)} <z ^ {(1)}, ldots, X ^ { (p)} <z ^ {(p)}) $, for everyone $ z = (z ^ {(1)}, ldots, z ^ {(p)}) in mathbb R ^ p $. Let $ X_1, ldots, X_n $ to be a sample of IID $ mathbb P $and for each coordinate $ j in {1, ldots, p } $, define the empirical mean $ overline {X ^ {(j)}}: = (1 / n) sum_ {i = 1} ^ nX_i ^ {(j)} in mathbb R $.

For fixed $ (z ^ {(1)}, ldots, z ^ {(p)}) in mathbb R ^ p $, what is $ lim_ {n rightarrow infty} mathbb P ( overline {X ^ {(1)}} <z ^ {(1)}, ldots, overline {X ^ {(p)}} <z ^ {(p)}) $ =?

My guess is $ lim_ {n rightarrow infty} mathbb P ( overline {X ^ {(1)}} <z ^ {(1)}, ldots, overline {X ^ {(p)}} <z ^ {(p)}) = F_X (z ^ {(1)}, ldots, z ^ {(p)}) $but I'm not sure. Could Glivenko-Cantelli's theorem lead to such a result?