On singular value distribution of large dimensional auto-covariance matrices

Let $(\varepsilon_j)_{j\geq 0}$ be a sequence of independent $p-$dimensional random vectors and $τ\geq1$ a given integer. From a sample $\varepsilon_1,\cdots,\varepsilon_{T+τ-1},\varepsilon_{T+τ}$ of the sequence, the so-called lag $-τ$ auto-covariance matrix is $C_τ=T^{-1}\sum_{j=1}^T\varepsilon_{τ+j}\varepsilon_{j}^t$. When the dimension $p$ is large compared to the sample size $T$, this paper establishes the limit of the singular value distribution of $C_τ$ assuming that $p$ and $T$ grow to infinity proportionally and the sequence satisfies a Lindeberg condition on fourth order moments. Compared to existing asymptotic results on sample covariance matrices developed in random matrix theory, the case of an auto-covariance matrix is much more involved due to the fact that the summands are dependent and the matrix $C_τ$ is not symmetric. Several new techniques are introduced for the derivation of the main theorem.

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