The Singular Values of the GOE

As a unifying framework for examining several properties that nominally involve eigenvalues, we present a particular structure of the singular values of the Gaussian orthogonal ensemble (GOE): the even-location singular values are distributed as the positive eigenvalues of a Gaussian ensemble with chiral unitary symmetry (anti-GUE), while the odd-location singular values, conditioned on the even-location ones, can be algebraically transformed into a set of independent $\chi$-distributed random variables. We discuss three applications of this structure: first, there is a pair of bidiagonal square matrices, whose singular values are jointly distributed as the even- and odd-location ones of the GOE; second, the magnitude of the determinant of the GOE is distributed as a product of simple independent random variables; third, on symmetric intervals, the gap probabilities of the GOE can be expressed in terms of the Laguerre unitary ensemble (LUE). We work specifically with matrices of finite order, but by passing to a large matrix limit, we also obtain new insight into asymptotic properties such as the central limit theorem of the determinant or the gap probabilities in the bulk-scaling limit. The analysis in this paper avoids much of the technical machinery (e.g. Pfaffians, skew-orthogonal polynomials, martingales, Meijer $G$-function, etc.) that was previously used to analyze some of the applications.Comment: Introduction extended, typos corrected, reference added. 31 pages, 1 figur

Universal microscopic correlation functions for products of independent Ginibre matrices

We consider the product of n complex non-Hermitian, independent random matrices, each of size NxN with independent identically distributed Gaussian entries (Ginibre matrices). The joint probability distribution of the complex eigenvalues of the product matrix is found to be given by a determinantal point process as in the case of a single Ginibre matrix, but with a more complicated weight given by a Meijer G-function depending on n. Using the method of orthogonal polynomials we compute all eigenvalue density correlation functions exactly for finite N and fixed n. They are given by the determinant of the corresponding kernel which we construct explicitly. In the large-N limit at fixed n we first determine the microscopic correlation functions in the bulk and at the edge of the spectrum. After unfolding they are identical to that of the Ginibre ensemble with n=1 and thus universal. In contrast the microscopic correlations we find at the origin differ for each n>1 and generalise the known Bessel-law in the complex plane for n=2 to a new hypergeometric kernel 0_F_n-1.Comment: 20 pages, v2 published version: typos corrected and references adde

Outage probability analysis for the multi-carrier NOMA downlink relying on statistical CSI

In this treatise, we derive tractable closed-form expressions for the outage probability of the single cell multi-carrier non-orthogonal multiple access (MC-NOMA) downlink, where the transmitter side only has statistical CSI knowledge. In particular, we analyze the outage probability with respect to the total data rates (summed over all subcarriers), given a minimum target rate for the individual users. The calculation of outage probability for the distant user is challenging, since the total rate expression is given by the sum of logarithmic functions of the ratio between two shifted exponential random variables, which are dependent. In order to derive the closed-form outage probability expressions both for two subcarriers and for a general case of multiple subcarriers, efficient approximations are proposed. The probability density function (PDF) of the product of shifted exponential distributions can be determined for the near user by the Mellin transform and the generalized upper incomplete Fox’s H function. Based on this PDF, the corresponding outage probability is presented. Finally, the accuracy of our outage analysis is verified by simulation results

Small sample confidence bands for the survival functions under proportional hazards model

In this work, a saddlepoint-based method is developed for generating small sample confidence bands for the population survival function from the Kaplan-Meier (KM), the product limit (PL), and Abdushukurov-Cheng-Lin (ACL) survival function estimators, under the proportional hazards model. In the process the exact distribution of these estimators is derived and developed mid-population tolerance bands for said estimators. The proposed saddlepoint method depends upon the Mellin transform of the zero-truncated survival estimator which is derived for the KM, PL, and ACL estimators. These transforms are inverted via saddlepoint approximations to yield highly accurate approximations to the cumulative distribution functions of the respective cumulative hazard function estimators and these distribution functions are then inverted to produce saddlepoint confidence bands. The saddlepoint confidence bands for the KM, PL and ACL estimators is compared with those obtained from competing large sample methods as well as those obtained from the exact distribution. In the simulation studies it is found that the saddlepoint confidence bands are very close to the confidence bands derived from the exact distribution, while being much easier to compute, and outperform the competing large sample methods in terms of coverage probability --Abstract, page iii

Involution factorizations of Ewens random permutations

An involution is a bijection that is its own inverse. Given a permutation $\sigma$ of $[n],$ let $\mathsf{invol}(\sigma)$ denote the number of ways to express $\sigma$ as a composition of two involutions of $[n].$ The statistic $\mathsf{invol}$ is asymptotically lognormal when the symmetric groups $\mathfrak{S}_n$ are each equipped with Ewens Sampling Formula probability measures of some fixed positive parameter $\theta.$ This paper strengthens and generalizes previously determined results on the limiting distribution of $\log(\mathsf{invol})$ for uniform random permutations, i.e. the specific case of $\theta = 1$. We also investigate the first two moments of $\mathsf{invol}$ itself.Comment: 23 pages, no figures. Some minor edits. Extra material added to sections 2 and 4 and concluding remark

Special Functions: Fractional Calculus and the Pathway for Entropy

Historically, the notion of entropy emerged in conceptually very distinct contexts. This book deals with the connection between entropy, probability, and fractional dynamics as they appeared, for example, in solar neutrino astrophysics since the 1970's (Mathai and Rathie 1975, Mathai and Pederzoli 1977, Mathai and Saxena 1978, Mathai, Saxena, and Haubold 2010). The original solar neutrino problem, experimentally and theoretically, was resolved through the discovery of neutrino oscillations and was recently enriched by neutrino entanglement entropy. To reconsider possible new physics of solar neutrinos, diffusion entropy analysis, utilizing Boltzmann entropy, and standard deviation analysis was undertaken with Super-Kamiokande solar neutrino data. This analysis revealed a non-Gaussian signal with harmonic content. The Hurst exponent is different from the scaling exponent of the probability density function and both Hurst exponent and scaling exponent of the Super-Kamiokande data deviate considerably from the value of ½, which indicates that the statistics of the underlying phenomenon is anomalous. Here experiment may provide guidance about the generalization of theory of Boltzmann statistical mechanics. Arguments in the so-called Boltzmann-Planck-Einstein discussion related to Planck's discovery of the black-body radiation law are recapitulated mathematically and statistically and emphasize from this discussion is pursued that a meaningful implementation of the complex ‘entropy-probability-dynamics’ may offer two ways for explaining the results of diffusion entropy analysis and standard deviation analysis. One way is to consider an anomalous diffusion process that needs to use the fractional space-time diffusion equation (Gorenflo and Mainardi) and the other way is to consider a generalized Boltzmann entropy by assuming a power law probability density function. Here new mathematical framework, invented by sheer thought, may provide guidance for the generalization of Boltzmann statistical mechanics. In this book Boltzmann entropy, generalized by Tsallis and Mathai, is considered. The second one contains a varying parameter that is used to construct an entropic pathway covering generalized type-1 beta, type-2 beta, and gamma families of densities. Similarly, pathways for respective distributions and differential equations can be developed. Mathai's entropy is optimized under various conditions reproducing the well-known Boltzmann distribution, Raleigh distribution, and other distributions used in physics. Properties of the entropy measure for the generalized entropy are examined. In this process the role of special functions of mathematical physics, particularly the H-function, is highlighted

Universal distribution of Lyapunov exponents for products of Ginibre matrices

Starting from exact analytical results on singular values and complex eigenvalues of products of independent Gaussian complex random $N\times N$ matrices also called Ginibre ensemble we rederive the Lyapunov exponents for an infinite product. We show that for a large number $t$ of product matrices the distribution of each Lyapunov exponent is normal and compute its $t$-dependent variance as well as corrections in a $1/t$ expansion. Originally Lyapunov exponents are defined for singular values of the product matrix that represents a linear time evolution. Surprisingly a similar construction for the moduli of the complex eigenvalues yields the very same exponents and normal distributions to leading order. We discuss a general mechanism for $2\times 2$ matrices why the singular values and the radii of complex eigenvalues collapse onto the same value in the large-$t$ limit. Thereby we rederive Newman's triangular law which has a simple interpretation as the radial density of complex eigenvalues in the circular law and study the commutativity of the two limits $t\to\infty$ and $N\to\infty$ on the global and the local scale. As a mathematical byproduct we show that a particular asymptotic expansion of a Meijer G-function with large index leads to a Gaussian.Comment: 36 pages, 6 figure