Random matrices by MA models and compound free Poisson laws (Q2874344)

It is shown that the limit spectral measure given by \textit{O. Pfaffel} and \textit{E. Schlemm} [Probab. Math. Stat. 31, No. 2, 313--329 (2011; Zbl 1260.62072)] is, in fact, a compound free Poisson law. More precisely, let \(X_n\) be the \(p \times n\) random matrix with dependence such that the \(i\)-th row of \(X_n\) is given by a linear process of the form NEWLINE\[NEWLINE ( X_{i,j} )_{j=1}^n = \left( \sum_{\ell = 0}^{\infty} c_{\ell} Z_{i, j- \ell} \right)_{j=1}^n \quad \text{with} \quad c_{\ell} \in {\mathbb R}, \tag{1} NEWLINE\]NEWLINE where the set \(\{ Z_{i,j} \}\) is given as the family of independent standardized (i.e., mean zero and variance one) random variables with uniformly bounded fourth moments and the Lindeberg-type condition. In the aforementioned paper, Pfaffel and Schlemm have investigated the Marchenko-Pastur type limit (\(n \to \infty\) and \(\lim_{ n \to \infty} n/p = \lambda > 0\)) of the sample covariance matrix \(W_n\) \(:=\) \(p^{-1} X_n \, ^tX_n\) which is a \(p \times p\) symmetric matrix, and have also determined the limit spectral measure \(\mu\) by giving the functional equation for its Stieltjes transform \(m_{\mu}\) NEWLINE\[NEWLINE m_{\mu} (z) := \int \frac{1}{ x - z} \mu( dx ), \quad \forall z \in {\mathbb C}^+. \tag{2} NEWLINE\]NEWLINE The present authors show not only that such a limit spectral measure \(\mu\) is a compound free Poisson law, but also that, in the special dependent case in terms of an MA (= moving average) modeled Gaussian process, the corresponding sample covariance matrix can be regarded as compound Wishart matrix. Finally, an application of the compound Wishart matrix to some statistical data analysis of time series is given as well. For other related works, see, e.g. [\textit{J.-S. Xie}, Stat. Probab. Lett. 83, No. 2, 543--550 (2013; Zbl 1267.60034)] for examples of the Marchenko-Pastur-type limit spectral distribution for normalized sample covariance matrices.