Amalgamated Free Lévy Processes as Limits of Sample Covariance Matrices

We prove the existence of joint limiting spectral distributions for families of random sample covariance matrices modeled on fluctuations of discretized Lévy processes. These models were first considered in applications of random matrix theory to financial data, where datasets exhibit both strong multicollinearity and non-normality. When the underlying Lévy process is non-Gaussian, we show that the limiting spectral distributions are distinct from Marčenko–Pastur. In the context of operator-valued free probability, it is shown that the algebras generated by these families are asymptotically free with amalgamation over the diagonal subalgebra. This framework is used to construct operator-valued $$^*$$ ∗ -probability spaces, where the limits of sample covariance matrices play the role of non-commutative Lévy processes whose increments are free with amalgamation.

A Method for the Reduction of Out-of-Band Measurement Errors in Multi-band Instruments using Synthetic Source Distributions

A method to reduce multi-band sensor measurement biases due to finite out-of-band response is described. The method takes advantage of the fact that out-of-band measurement errors cancel if the calibration source and the measured source have the same spectral distributions—independent of their spectral distributions or the magnitude of a sensor band’s out-of-band response. Using a known spectral responsivity, a synthetic, arbitrary source spectral distribution can replace a realized spectral distribution in the measurement equation and the signal can be calculated rather than measured. Given the freedom to select any arbitrary distribution for the synthetic source, the efficacy of the approach depends on the fidelity of the replication of the measured spectrum by the synthetic source spectrum. To illustrate the method, an example application is given of top-of-the-atmosphere measurements of water-leaving radiance by multi-band filter radiometers on celestial Earth-viewing sensors.

A study of the influence of copper sulfate on the spectral properties of common buckwheat (Fagopyrum esculentum)

The influence of pollutants on the spectral properties of common buckwheat (Fagopyrum esculentum) has been investigated insufficiently, compared to the cereals from the Poaceae family. A two-stage spectral survey has been carried out, growing common buckwheat in containers with set concentrations of copper(II) sulfate in soil both in laboratory conditions and in the open air. Spectral distributions of diffuse reflectance of the plants were registered in the range of 400 – 1100 nm, and spectral indices were calculated, using wavelengths corresponding to spectral features of plant pigments. Simultaneously, digital photos were taken to account for projective cover of the plants. Four spectral indices were found to depend quantitatively on pollutant concentration, three of them taking extreme values at the time of maximal projective cover. When growing buckwheat in the open air, lower temperatures and higher irradiation lead to closer values of spectral indices corresponding to different copper concentration, than in laboratory conditions. The results show the usability of common buckwheat as an indicator of soil pollution by copper.

Upwelling Irradiance below Sea Ice—PAR Intensities and Spectral Distributions

Upwelling and downwelling spectral (320–920 nm) distributions and photosynthetic active radiation (PAR) intensities were measured below a first-year land-fast sea ice in a western Greenland fjord with and without a snow cover. Time-series of surface upwelling PAR, downwelling PAR, and under-ice PAR were also obtained. Spectral distributions of upwelling and downwelling irradiances were similar except for reduced intensities in the UV, the red, and NIR parts of the spectrum when the ice was snow-covered. Upwelling PAR amounted to about 10% of downwelling intensities, giving 5.1 µmol photons m−2 s−1 at the bottom of the ice with a snow cover and 8.2 µmol photons m−2 s−1 without. PAR partitioning analyses showed that the upwelling was related to scattering by suspended particles in the water column. A snow melt increased under-ice daily maximum downwelling PAR from 50 to 180 µmol photons m−2 s−1 and overall under-ice PAR of 55 and 198 µmol photons m−2 s−1 with 10% upwelling. It is concluded that upwelling PAR below sea ice might be an important factor regarding sea ice algae photophysiology and performance with a 10% higher PAR; specifically when PAR > Ek the light saturation point of the sea ice algae.

A method to measure the spectral responsivity of a photometer using optical excitations with arbitrary spectral distributions

The spectral responsivity of a photometer is usually measured using very narrow optical excitations, provided by a monochromator or a tuneable laser. This article describes a technique to measure the spectral responsivity using an arbitrary number of optical excitations having any type of spectral distribution. The problem is formulated as an inverse problem which is solved using a probabilistic approach based on Bayes’ theorem. The method requires a prior knowledge of the spectral responsivity, which can be proportional to the standard photopic function, with an uncertainty level related to the spectral match index of the photometer. Using this method, the estimation can be performed from data provided by a simple experimental set-up. The numerical application provides a stable and unique solution to the inverse problem, along with the estimation uncertainties. Using a tuneable LED source, the method was applied to an illuminance measurement head, giving an estimation of its spectral responsivity from 380 to 780 nm with a step of 1 nm. The results were in good agreement with data obtained by a monochromator-based technique. Our measurement had larger uncertainties towards the red and blue limits of the spectrum as the light source provided very little light at these wavelengths.

High-dimensional regimes of non-stationary Gaussian correlated Wishart matrices

We study the high-dimensional asymptotic regimes of correlated Wishart matrices [Formula: see text], where [Formula: see text] is a [Formula: see text] Gaussian random matrix with correlated and non-stationary entries. We prove that under different normalizations, two distinct regimes emerge as both [Formula: see text] and [Formula: see text] grow to infinity. The first regime is the one of central convergence, where the law of the properly renormalized Wishart matrices becomes close in Wasserstein distance to that of a Gaussian orthogonal ensemble matrix. In the second regime, a non-central convergence happens, and the law of the normalized Wishart matrices becomes close in Wasserstein distance to that of the so-called Rosenblatt–Wishart matrix recently introduced by Nourdin and Zheng. We then proceed to show that the convergences stated above also hold in a functional setting, namely as weak convergence in [Formula: see text]. As an application of our main result (in the central convergence regime), we show that it can be used to prove convergence in expectation of the empirical spectral distributions of the Wishart matrices to the semicircular law. Our findings complement and extend a rich collection of results on the study of the fluctuations of Gaussian Wishart matrices, and we provide explicit examples based on Gaussian entries given by normalized increments of a bi-fractional or a sub-fractional Brownian motion.

High Order Coherence Functions and Spectral Distributions as Given by the Scully-Lamb Quantum Theory of the Laser

We propose and demonstrate a method for measuring the time evolution of the off-diagonal elements ρn,n+k(t) of the reduced density matrix obtained from the quantum theory of the laser. The decay rates of the off-diagonal matrix element ρn,n+k(t) (k = 2,3) are measured for the first time and compared with that of ρn,n+1(t), which corresponds to the linewidth of the laser. The experimental results agree with the Scully-Lamb quantum theory of the laser.