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Mathematical statistics --- Asymptotic theory.

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We examine the implications of arbitrage in a market with many assets. The absence of arbitrage opportunities implies that the linear functionals that give the mean and cost of a portfolio are continuous; hence there exist unique portfolios that represent these functionals. The mean variance efficient set is a cone generated by these portfolios. Ross [16, 18J showed that if there is a factor structure, then the distance between the vector or mean returns and the space spanned by the factor loadings is bounded as the number of assets increases. We show that if the covariance matrix of asset returns has only K unbounded eigenvalues, then the corresponding K eigenvectors converge and play the role of factor loadings in Ross' result. Hence only a principal components analysis is needed to test the arbitrage pricing theory. Our eigenvalue conditional can hold even though conventional measures of the approximation error in a K factor model are unbounded. We also resolve the question of when a market with many assets permits so much diversification that risk-free investment opportunities are available.

Mathematical statistics --- Asymptotic theory.

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Partial differential equations (PDEs) describe technological phenomena and processes used for the analysis, design, and modeling of technical products. Solutions of spatial and transient PDEs are realized by using the PDE Toolbox included in the MATLAB® software. MATLAB® is introduced here as an essential foundation for PDE, and the Modeler of the PDE Toolbox, with appropriate explanatory solutions, is applied to engineering problems in mechanics, heat/mass transfer, tribology, materials science, physics, and biotechnology. The appendixes contain collections of commands and functions used to solve actual engineering problems.

Differential equations, Partial --- Asymptotic theory

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Mathematical statistics --- Probabilities. --- Asymptotic theory.

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Partial differential equations --- Dirichlet problem --- -Boundary value problems --- Asymptotic theory --- -Asymptotic theory --- Dirichlet, Problème de. --- Asymptotic theory. --- Asymptotic theory in the Dirichlet problem --- Asymptotic expansions