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 p­variate Wishart distribution on n degrees of freedom with identity covariance. Consider the limit of large p and n with $n/p = \gamma \ge 1$. When centered by $\mu_p = (\sqrt{n-1} + \sqrt{p})^2$ and scaled by $\sigma_p = (\sqrt{n-1} + \sqrt{p})(1/\sqrt{n-1} + 1/\sqrt{p}^{1/3}$, the distribution of x(1) approaches the Tracey-Widom law of order 1, which is defined in terms of the Painleve II differential equation and can be numerically evaluated and tabulated in software. Simulations show the 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.

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On the distribution of the largest eigenvalue in principal components analysis

Functions of one complex variable II

Singular perturbation methods, such as the method of multiple scales and the method of matched asymptotic expansions, give series in a small parameter e which are asymptotic but (usually) divergent. In this survey, we use a plethora of examples to illustrate the cause of the divergence, and explain how this knowledge can be exploited to generate a 'hyperasymptotic' approximation. This adds a second asymptotic expansion, with different scaling assumptions about the size of various terms in the problem, to achieve a minimum error much smaller than the best possible with the original asymptotic series. (This rescale-and-add process can be repeated further.) Weakly nonlocal solitary waves are used as an illustration.

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The Devil's Invention: Asymptotic, Superasymptotic and Hyperasymptotic Series

Let A and B be independent, central Wishart matrices in p variables with common covariance and having m and n degrees of freedom, respectively. The distribution of the largest eigenvalue of (A + B)(-1)B has numerous applications in multivariate statistics, but is difficult to calculate exactly. Suppose that m and n grow in proportion to p. We show that after centering and, scaling, the distribution is approximated to second-order, O(p(-2/3)), by the Tracy-Widom law. The results are obtained for both complex and then real-valued data by using methods of random matrix theory to study the largest eigenvalue of the Jacobi unitary and orthogonal ensembles. Asymptotic approximations of Jacobi polynomials near the largest zero play a central role.

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Multivariate analysis and jacobi ensembles: largest eigenvalue, tracy-widom limits and rates of convergence.

For non-Hermitian Hamiltonians with an isolated degeneracy (‘exceptional point’), a model for cycling around loops that enclose or exclude the degeneracy is solved exactly in terms of Bessel functions. Floquet solutions, returning exactly to their initial states (up to a constant) are found, as well as exact expressions for the adiabatic multipliers when the evolving states are represented as a superposition of eigenstates of the instantaneous Hamiltonian. Adiabatically (i.e. for slow cycles), the multipliers of exponentially subdominant eigenstates can vary wildly, unlike those driven by Hermitian operators, which change little. These variations are explained as an example of the Stokes phenomenon of asymptotics. Improved (superadiabatic) approximations tame the variations of the multipliers but do not eliminate them.

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Slow non-Hermitian cycling: exact solutions and the Stokes phenomenon

Bessel functions of purely imaginary order are examined. Solutions of both the modified and unmodified Bessel equations are defined which, when their order is purely imaginary and their argument is real and positive, are pairs of real numerically satisfactory functions. Recurrence relations, analytic continuation formulas, power series representations, Wronskian relations, integral representations, behavior at singularities, and asymptotic forms of the zeros are derived for these numerically satisfactory functions. Also, asymptotic expansions in terms of elementary and Airy functions are derived for the Bessel functions when their order is purely imaginary and of large absolute value.Second-order linear ordinary differential equations having a large parameter and a simple pole are then examined, for the case where the exponent of the pole is complex. Asymptotic expansions are derived for the solutions in terms of the numerically satisfactory Bessel functions of purely imaginary order.

Bessel functions of purely imaginary order, with an application to second-order linear differential equations having a large parameter

A class of second-order linear differential equations with a large parameter u is considered. It is shown that Liouville-Green type expansions for solutions can be expressed using factorial series in the parameter, and that such expansions converge for Re (u) > 0, uniformly for the independent variable lying in a certain subdomain of the domain of asymptotic validity. The theory is then applied to obtain convergent expansions for modified Bessel functions of large order.

Convergent Liouville-Green expansions for second-order linear differential equations, with an application to Bessel functions

The asymptotic behavior, as $\kappa \to \infty $, of the Whittaker confluent hypergeometric functions $M_{\kappa ,\mu } (z)$ and $W_{\kappa ,\mu } (z)$ is examined. Asymptotic expansions are derived in terms of Bessel and Airy functions, the results being uniformly valid for real values of $\kappa $ and $\mu $ such that ${{0 \leqq \mu } / {\kappa \leqq A < 1}}$ (A an arbitrary constant), and for all complex values of the argument z. Explicit error bounds are available for all the approximations.

Uniform asymptotic expansions for Whittaker's confluent hypergeometric functions

Uniform asymptotic expansions are derived for conical functions, Legendre functions of order µ and degree −½ + iτ, where µ and τ are non-negative real parameters. As τ → ∞, expansions are furnished for the conical functions which involve Bessel functions of order µ. These expansions are uniformly valid for 0 ≦ µ ≦ Aτ (A an arbitrary positive constant), and are also uniformly valid for Re (z) ≧ 0 in the complex argument case, and 0 ≦ z < ∞ in the real argument case. The case µ → ∞ is also considered, and expansions are furnished which are uniformly valid in the same z regions for 0 ≦ τ ≧ Bµ (B an arbitrary positive constant); in the cases where Re(z) ≧ 0 and 1 ≦ z < ∞, the expansions involve Bessel functions of purely imaginary order iτ, and in the case where 0 ≦ z < 1 the expansions involve elementary functions.

Conical functions with one or both parameters large

Uniform asymptotic expansions are derived for solutions of the differential equation d 2 W/dζ 2 = ( u 2 ζ 2 + βu + ψ( u , ζ))W, which are uniformly valid for u real and large, β bounded (real or complex), and £ lying in a well-defined bounded or unbounded complex domain, which contains the origin. The function ψ(u, ζ) is assumed to be holomorphic in this domain, and is o ( u / ln( u )) uniformly as u —> oo. The approximations involve parabolic cylinder functions, and include explicit and realistic error bounds. The new theory is then applied to the complementary incomplete gamma function Γ( α , α x ), furnishing an asymptotic approximation which is uniformly valid for x lying in a complex domain which properly contains all x satisfying |arg(x)| oo.

Asymptotic Solutions of Second-Order Linear Differential Equations having Almost Coalescent Turning Points, with an Application to the Incomplete Gamma Function

T. M. Dunster

Papers

Bessel functions of purely imaginary order, with an application to second-order linear differential equations having a large parameter

Convergent Liouville-Green expansions for second-order linear differential equations, with an application to Bessel functions

Uniform asymptotic expansions for Whittaker's confluent hypergeometric functions

Conical functions with one or both parameters large

Asymptotic Solutions of Second-Order Linear Differential Equations having Almost Coalescent Turning Points, with an Application to the Incomplete Gamma Function