We investigate the dynamic behavior of the stationary random process defined by a central complex Wishart matrix @math as it varies along a certain dimension @math . We characterize the second-order joint cumulative distribution function (cdf) of the largest eigenvalue, and the second-order joint cdf of the smallest eigenvalue of this matrix. We show that both cdfs can be expressed in exact closed-form in terms of a finite number of well-known special functions in the context of communication theory. As a direct application, we investigate the dynamic behavior of the parallel channels associated with multiple-input multiple-output (MIMO) systems in the presence of Rayleigh fading. Studying the complex random matrix that defines the MIMO channel, we characterize the second-order joint cdf of the signal-to-noise ratio (SNR) for the best and worst channels. We use these results to study the rate of change of MIMO parallel channels, using different performance metrics. For a given value of the MIMO channel correlation coefficient, we observe how the SNR associated with the best parallel channel changes slower than the SNR of the worst channel. This different dynamic behavior is much more appreciable when the number of transmit ( @math ) and receive ( @math ) antennas is similar. However, as @math is increased while keeping @math fixed, we see how the best and worst channels tend to have a similar rate of change.
Cellular immune control of HIV is mediated, in part, by induction of single amino acid mutations that reduce viral fitness, but compensatory mutations limit this effect. Here, we sought to determine if higher order constraints on viral evolution exist, because some coordinately linked combinations of mutations may hurt viability. Immune targeting of multiple sites in such a multidimensionally conserved region might render the virus particularly vulnerable, because viable escape pathways would be greatly restricted. We analyzed available HIV sequences using a method from physics to reveal distinct groups of amino acids whose mutations are collectively coordinated (“HIV sectors”). From the standpoint of mutations at individual sites, one such group in Gag is as conserved as other collectively coevolving groups of sites in Gag. However, it exhibits higher order conservation indicating constraints on the viability of viral strains with multiple mutations. Mapping amino acids from this group onto protein structures shows that combined mutations likely destabilize multiprotein structural interactions critical for viral function. Persons who durably control HIV without medications preferentially target the sector in Gag predicted to be most vulnerable. By sequencing circulating viruses from these individuals, we find that individual mutations occur with similar frequency in this sector as in other targeted Gag sectors. However, multiple mutations within this sector are very rare, indicating previously unrecognized multidimensional constraints on HIV evolution. Targeting such regions with higher order evolutionary constraints provides a novel approach to immunogen design for a vaccine against HIV and other rapidly mutating viruses. I INTRODUCTION II PROPERTIES OF RANDOM MATRIX THEORY III APPLICATIONS OF RANDOM MATRIX THEORY The Wishart Autoregressive (WAR) process is a dynamic model for time series of multivariate stochastic volatility. The WAR naturally accommodates the positivity and symmetry of volatility matrices and provides closed-form non-linear forecasts. The estimation of the WAR is straighforward, as it relies on standard methods such as the Method of Moments and Maximum Likelihood. For illustration, the WAR is applied to a sequence of intraday realized volatility-covolatility matrices from the Toronto Stock MarketÂ (TSX). We study the realized variance of sample minimum variance portfolios of arbitrarily high dimension. We consider the use of covariance matrix estimators based on shrinkage and weighted sampling. For such improved portfolio implementations, the otherwise intractable problem of characterizing the realized variance is tackled here by analyzing the asymptotic convergence of the risk measure. Rather than relying on less insightful classical asymptotics, we manage to deliver results in a practically more meaningful limiting regime, where the number of assets remains comparable in magnitude to the sample size. Under this framework, we provide accurate estimates of the portfolio realized risk in terms of the model parameters and the underlying investment scenario, i.e., the unknown asset return covariance structure. In-sample approximations in terms of only the available data observations are known to considerably underestimate the realized portfolio risk. If not corrected, these deviations might lead in practice to inaccurate and overly optimistic investment decisions. Therefore, along with the asymptotic analysis, we also provide a generalized consistent estimator of the out-of-sample portfolio variance that only depends on the set of observed returns. Based on this estimator, the model free parameters, i.e., the sample weighting coefficients and the shrinkage intensity defining the minimum variance portfolio implementation, can be optimized so as to minimize the realized variance while taken into account the effect of estimation risk. Our results are based on recent contributions in the field of random matrix theory. Numerical simulations based on both synthetic and real market data validate our theoretical findings under a non-asymptotic, finite-dimensional setting. Finally, our proposed portfolio estimator is shown to consistently outperform a widely applied benchmark implementation. Abstract We review the development of random-matrix theory (RMT) during the last fifteen years. We emphasize both the theoretical aspects, and the application of the theory to a number of fields. These comprise chaotic and disordered systems, the localization problem, many-body quantum systems, the Calogero-Sutherland model, chiral symmetry breaking in QCD, and quantum gravity in two dimensions. The review is preceded by a brief historical survey of the developments of RMT and of localization theory since their inception. We emphasize the concepts common to the above-mentioned fields as well as the great diversity of RMT. In view of the universality of RMT, we suggest that the current development signals the emergence of a new “statistical mechanics”: Stochasticity and general symmetry requirements lead to universal laws not based on dynamical principles. Center for Neur obiology and Behavior , Columbia University ,,Colle ge of Physicians and Sur geons, New York, Ne w York 10032, USA(Receive d 18 July 2006; published 2 Nov ember 2006)The dynamics of neural networks is insuenced strongly by the spectrum of eigen values of the matrixdescribing their synaptic connectivity .In large networks, elements of the synaptic connecti vity matrix canbe chosen randomly from appropriate distributions, making results from random matrix theory highlyrele vant. Unfortunately , classic results on the eigen value spectra of random matrices do not apply tosynaptic connectivity matrices because of the constraint that indi vidual neurons are either ex citatory orinhibitory . Therefore, we compute eigen value spectra of large random matrices with ex citatory andinhibitory columns drawn from distributions with different means and equal or different variances. 