  We derive the Marchenko-Pastur (MP) law for sample covariance matrices of the
form $V_n=\frac{1}{n}XX^T$, where $X$ is a $p\times n$ data matrix and $p/n\to
y\in(0,\infty)$ as $n,p \to \infty$. We assume the data in $X$ stems from a
correlated joint normal distribution. In particular, the correlation acts both
across rows and across columns of $X$, and we do not assume a specific
correlation structure, such as separable dependencies. Instead, we assume that
correlations converge uniformly to zero at a speed of $a_n/n$, where $a_n$ may
grow mildly to infinity. We employ the method of moments tightly: We identify
the exact condition on the growth of $a_n$ which will guarantee that the
moments of the empirical spectral distributions (ESDs) converge to the MP
moments. If the condition is not met, we can construct an ensemble for which
all but finitely many moments of the ESDs diverge. We also investigate the
operator norm of $V_n$ under a uniform correlation bound of $C/n^{\delta}$,
where $C,\delta>0$ are fixed, and observe a phase transition at $\delta=1$. In
particular, convergence of the operator norm to the maximum of the support of
the MP distribution can only be guaranteed if $\delta>1$. The analysis leads to
an example for which the MP law holds almost surely, but the operator norm
remains stochastic in the limit, and we provide its exact limiting
distribution.
