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Let x �1 � denote the square of the largest singular value of an n × p matrix X, all of whose entries are independent standard Gaussian variates. Equivalently, x �1 � is the largest principal component variance of the covariance matrix X ′ X, or the largest eigenvalue of a p-variate Wishart distribution on n degrees of freedom with identity covariance. Consider the limit of large p and n with n/p = γ ≥ 1. When centered by µ p = � √ n − 1 + √ p � 2 and scaled by σ p = � √ n − 1 + √ p��1 / √ n − 1 + 1 / √ p � 1/3 � the distribution of x �1 � approaches the Tracy–Widom lawof order 1, which is defined in terms of the Painlevé II differential equation and can be numerically evaluated and tabulated in software. Simulations showthe approximation to be informative for n and p as small as 5. The limit is derived via a corresponding result for complex Wishart matrices using methods from random matrix theory. The result suggests that some aspects of large p multivariate distribution theory may be easier to apply in practice than their fixed p counterparts. 1. Introduction. The

Let B... In this paper we prove that, under certain conditions on the eigenvalues of Tn , for any closed interval outside the support of the limit, with probability 1 there will be no eigenvalues in this interval for all n sufficiently large.

We consider the asymptotic fluctuation behavior of the largest eigenvalue of certain sample covariance matrices in the asymptotic regime where both dimensions of the corresponding data matrix go to infinity. More precisely, let X be an n × p matrix, and let its rows be i.i.d. complex normal vectors with mean 0 and covariance �p. We show that for a large class of covariance matrices �p, the largest eigenvalue of X ∗ X is asymptotically distributed (after recentering and rescaling) as the Tracy–Widom distribution that appears in the study of the Gaussian unitary ensemble. We give explicit formulas for the centering and scaling sequences that are easy to implement and involve only the spectral distribution of the population covariance, n and p. The main theorem applies to a number of covariance models found in applications. For example, well-behaved Toeplitz matrices as well as covariance matrices whose spectral distribution is a sum of atoms (under some conditions on the mass of the atoms) are among the models the theorem can handle. Generalizations of the theorem to certain spiked versions of our models and a.s. results about the largest eigenvalue are given. We also discuss a simple corollary that does not require normality of the entries of the data matrix and some consequences for applications in multivariate statistics.

Let Xn be n×N containing i.i.d. complex entries and unit variance (sum of variances of real and imaginary parts equals 1), σ>0 constant, and Rn an n × N random matrix independent of Xn. Assume, almost surely, as n →∞, the empirical distribution function (e.d.f.) of the eigenvalues of 1 N RnR ∗ n converges in distribution to a nonrandom probability distribution function (p.d.f.), and the ratio n N tends to a positive number. Then it is shown that, almost surely, the e.d.f. of the eigenvalues of 1 N (Rn + σXn)(Rn + σXn) ∗ converges in distribution. The limit is nonrandom and is characterized in terms of its Stieltjes transform, which satisfies a certain equation. 1.

This paper presents a set of algorithms for the creation of underwater mosaics and illustrates their use as visual maps for underwater vehicle navigation. First, we describe the automatic creation of video mosaics, which deals with the problem of image motion estimation in a robust and automatic way. The motion estimation is based on a initial matching of corresponding areas over pairs of images, followed by the use of a robust matching technique, which can cope with a high percentage of incorrect matches. Several motion models, established under the projective geometry framework, allow for the creation of high quality mosaics where no assumptions are made about the camera motion. Several tests were run on underwater image sequences, testifying to the good performance of the implemented matching and registration methods. Next, we deal with the issue of determining the 3D position and orientation of a vehicle from new views of a previously created mosaic. The problem of pose estimation is tackled, using the available information on the camera intrinsic parameters. This information ranges from the full knowledge to the case where they are estimated using a self-calibration technique based on the analysis of an image sequence captured under pure rotation. The performance of the 3D positioning algorithms is evaluated using images for which accurate ground truth is available. c ○ 2000 Academic Press

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In this paper, we study the problem of learning phylogenies and hidden Markov models. We call the Markov model nonsingular if all transtion matrices have determinants bounded away from 0 (and 1). We highlight the role of the nonsingularity condition for the learning problem. Learning hidden Markov models without the nonsingularity condition is at least as hard as learning parity with noise. On the other hand, we give a polynomial-time algorithm for learning nonsingular phylogenies and hidden Markov models.

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Abstract. We consider a Markovian multiserver queueing model with time dependent parameters where waiting customers may abandon and subsequently retry. We provide simple fluid and diffusion approximations to estimate the mean, variance, and density for both the queue length and virtual waiting time processes arising in this model. These approximations, which are generated by numerically integrating only 7 ordinary differential equations, are justified by limit theorems where the arrival rate and number of servers grow large. We compare our approximations to simulations, and they perform extremely well.