This article is concerned with the spectral behavior of $p$-dimensional linear processes in the moderately high-dimensional case when both dimensionality $p$ and sample size $n$ tend to infinity so that $p/nto0$. It is shown that, under an appropriate set of assumptions, the empirical spectral distributions of the renormalized and symmetrized sample autocovariance matrices converge almost surely to a nonrandom limit distribution supported on the real line. The key assumption is that the linear process is driven by a sequence of $p$-dimensional real or complex random vectors with i.i.d. entries possessing zero mean, unit variance and finite fourth moments, and that the $ptimes p$ linear process coefficient matrices are Hermitian and simultaneously diagonalizable. Several relaxations of these assumptions are discussed. The results put forth in this paper can help facilitate inference on model parameters, model diagnostics and prediction of future values of the linear process.
Scanning tunneling microscopy and spectroscopy have been used to investigate the femtosecond dynamics of Dirac fermions in the topological insulator Bi$_2$Se$_3$ ultrathin films. At two-dimensional limit, bulk electrons becomes quantized and the quantization can be controlled by film thickness at single quintuple layer level. By studying the spatial decay of standing waves (quasiparticle interference patterns) off steps, we measure directly the energy and film thickness dependence of phase relaxation length $l_{phi}$ and inelastic scattering lifetime $tau$ of topological surface-state electrons. We find that $tau$ exhibits a remarkable $(E-E_F)^{-2}$ energy dependence and increases with film thickness. We show that the features revealed are typical for electron-electron scattering between surface and bulk states.
