Random Matrices: Theory and Applications

We use free probability to compute the limiting spectral properties of the harmonic mean of $n$ i.i.d. Wishart random matrices $W_i$ whose limiting aspect ratio is $\gamma \in (0,1)$ when $𝔼[W_i]=I$. We demonstrate an interesting phenomenon where the harmonic mean $H$ of the $n$ Wishart matrices is closer in operator norm to $𝔼[W_i]$ than the arithmetic mean $A$ for small $n$, after which the arithmetic mean is closer. We also prove some results for the general case where the expectation of the Wishart matrices are not the identity matrix.

We present a definition for second-order freeness in the quaternionic case. We demonstrate that this definition on a second-order probability space is asymptotically satisfied by independent symplectically invariant quaternionic matrices. This definition is different from the natural definition for complex and real second-order probability spaces, those motivated by the asymptotic behavior of unitarily invariant and orthogonally invariant random matrices respectively. Most notably, because the quaternionic trace does not have the cyclic property of a trace over a commutative field, the asymmetries which appear in the multi-matrix context result in an asymmetric contribution from the terms which appear symmetrically in the complex and real cases.

We consider properties of the ground state density for the $d$-dimensional Fermi gas in an harmonic trap. Previous work has shown that the $d$-dimensional Fourier transform has a very simple functional form. It is shown that this fact can be used to deduce that the density itself satisfies a third-order linear differential equation, previously known in the literature but from other considerations. It is shown too how this implies a closed form expression for the $2k$th non-negative integer moments of the density, and a second-order recurrence. Both can be extended to general Re$k>-d/2$. The moments, and the smoothed density, permit expansions in $1/{\tilde{M}}^2$, where $\tilde{M}=M+(d+1)/2$, with $M$ denoting the shell label. The moment expansion substituted in the second-order recurrence gives a generalization of the Harer–Zagier recurrence, satisfied by the coefficients of the $1/N^2$ expansion of the moments of the spectral density for the Gaussian unitary ensemble in random matrix theory.

In this note, we give a combinatorial and noncomputational proof of the asymptotics of the integer moments of the moments of the characteristic polynomials of Haar distributed unitary matrices as the size of the matrix goes to infinity. This is achieved by relating these quantities to a lattice point count problem. Our main result is a new explicit expression for the leading order coefficient in the asymptotic as a volume of a certain region involving continuous Gelfand–Tsetlin patterns with constraints.

We provide a generalization and new proofs of the formulas of Collins et al. for the spectrum of polynomials in cyclic monotone elements. This is applied to Random Matrices with discrete spectrum.

The asymptotic behavior 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 that the distribution behaves as a deterministic probability measure whose support ${𝒮}$ is characterized. It is also established that the squared singular values are almost surely located in a neighborhood of ${𝒮}$.

In this paper, we derive the Tracy–Widom law for the largest eigenvalue of sample covariance matrix generated by the vector autoregressive moving average model when the dimension is comparable to the sample size. This result is applied to make inference on the vector autoregressive moving average model. Simulations are conducted to demonstrate the finite sample performance of our inference.

In this text, based on elementary computations, we provide a perturbative expansion of the coordinates of the eigenvectors of a Hermitian matrix of large size perturbed by a random matrix with small operator norm whose entries in the eigenvector basis of the first one are independent, centered, with a variance profile. This is done through a perturbative expansion of spectral measures associated to the state defined by a given vector.

In order to further improve the efficiency of parameter estimation and reduce the number of estimated parameters, we adopt dimension reduction ideas of partial envelope model proposed by [Su and Cook, Partial envelopes for efficient estimation in multivariate linear regression, Biometrika 98 (2011) 133–146.] to center on some predictors of special interest. Based on the research results of [Cook et al., Envelopes and reduced-rank regression, Biometrika 102 (2015) 439–456.], we combine partial envelopes with reduced-rank regression to form reduced-rank partial envelope model which can reduce dimension efficiently. This method has the potential to perform better than both. Further, we demonstrate maximum likelihood estimators for the reduced-rank partial envelope model parameters, and exhibit asymptotic distribution and theoretical properties under normality. Meanwhile, we show selections of rank and partial envelope dimension. At last, under the normal and non-normal error distributions, simulation studies are carried out to compare our proposed reduced-rank partial envelope model with the other four methods, including ordinary least squares, reduced-rank regression, partial envelope model and reduced-rank envelope model. A real data analysis is also given to support the theoretic claims. The reduced-rank partial envelope estimators have shown promising performance in extensive simulation studies and real data analysis.

We focus on regression problems in which the predictors are naturally in the form of matrices. Reduced rank regression and related regularized method have been adapted to matrix regression. However, linear methods are restrictive in their expressive power. In this work, we consider a class of semiparametric additive models based on series estimation of nonlinear functions which interestingly induces a problem of 3rd order tensor regression with transformed predictors. Risk bounds for the estimator are derived and some simulation results are presented to illustrate the performances of the proposed method.