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 $\tau\geq1$ a given integer. From a sample $\varepsilon_1,\cdots,\varepsilon_{T+\tau-1},\varepsilon_{T+\tau}$ of the sequence, the so-called lag $-\tau$ auto-covariance matrix is $C_{\tau}=T^{-1}\sum_{j=1}^T\varepsilon_{\tau+j}\varepsilon_{j}^t$. When the dimension $p$ is large compared to the sample size $T$, this paper establishes the limit of the singular value distribution of $C_\tau$ 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 $C_\tau$ is not symmetric. Several new techniques are introduced for the derivation of the main theorem

On singular values distribution of a large auto-covariance matrix in the ultra-dimensional regime

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.Comment: 32 pages, 2 figure

Identifying the number of factors from singular values of a large sample auto-covariance matrix

Identifying the number of factors in a high-dimensional factor model has attracted much attention in recent years and a general solution to the problem is still lacking. A promising ratio estimator based on the singular values of the lagged autocovariance matrix has been recently proposed in the literature and is shown to have a good performance under some specific assumption on the strength of the factors. Inspired by this ratio estimator and as a first main contribution, this paper proposes a complete theory of such sample singular values for both the factor part and the noise part under the large-dimensional scheme where the dimension and the sample size proportionally grow to infinity. In particular, we provide the exact description of the phase transition phenomenon that determines whether a factor is strong enough to be detected with the observed sample singular values. Based on these findings and as a second main contribution of the paper, we propose a new estimator of the number of factors which is strongly consistent for the detection of all significant factors (which are the only theoretically detectable ones). In particular, factors are assumed to have the minimum strength above the phase transition boundary which is of the order of a constant; they are thus not required to grow to infinity together with the dimension (as assumed in most of the existing papers on high-dimensional factor models). Empirical Monte-Carlo study as well as the analysis of stock returns data attest a very good performance of the proposed estimator. In all the tested cases, the new estimator largely outperforms the existing estimator using the same ratios of singular values.Comment: This is a largely revised version of the previous manuscript (v1 & v2

Moment approach for singular values distribution of a large auto-covariance matrix

Let (εt)t>0(εt)t>0 be a sequence of independent real random vectors of pp-dimension and let XT=∑s+Tt=s+1εtε∗t−s/TXT=∑t=s+1s+Tεtεt−s∗/T be the lag-ss (ss is a fixed positive integer) auto-covariance matrix of εtεt. Since XTXT is not symmetric, we consider its singular values, which are the square roots of the eigenvalues of XTX∗TXTXT∗. Using the method of moments, we are able to investigate the limiting behaviors of the eigenvalues of XTX∗TXTXT∗ in two aspects. First, we show that the empirical spectral distribution of its eigenvalues converges to a nonrandom limit FF, which is a result previously developed in (J. Multivariate Anal. 137 (2015) 119–140) using the Stieltjes transform method. Second, we establish the convergence of its largest eigenvalue to the right edge of FF.published_or_final_versio

Comportement asymptotique de grandes matrices d'autocovariance entre passé et futur

The asymptotic behaviour of the distribution of the squared singular values of the sample autocovariance matrix between the past and the future of a high-dimensional complex Gaussian uncorrelated sequence is studied. Using Gaussian tools, it is established the distribution behaves as a deterministic probability measure whose support S is characterized. It is also established that the singular values to the square are almost surely located in a neighbourhood of S