  Let $(\varepsilon_{t})_{t>0}$ be a sequence of independent real random
vectors of $p$-dimension and let
$X_T=\sum_{t=s+1}^{s+T}\varepsilon_t\varepsilon^T_{t-s}/T$ be the lag-$s$ ($s$
is a fixed positive integer) auto-covariance matrix of $\varepsilon_t$. This
paper investigates the limiting behavior of the singular values of $X_T$ under
the so-called {\em ultra-dimensional regime} where $p\to\infty$ and
$T\to\infty$ in a related way such that $p/T\to 0$. First, we show that the
singular value distribution of $X_T$ after a suitable normalization converges
to a nonrandom limit $G$ (quarter law) under the forth-moment condition.
Second, we establish the convergence of its largest singular value to the right
edge of $G$. Both results are derived using the moment method.
