Sturm's Method in Counting Roots of Random Polynomial Equations

The problem of finding the probability distribution of the number of zeros in some real interval of a random polynomial whose coefficients have a given continuous joint density function is considered. An algorithm which enables one to express this probability as a multiple integral is presented. Formulas for the number of zeros of random quadratic polynomials and random polynomials of higher order, some coefficients of which are non-random and equal to zero, are derived via use of the algorithm. Finally, the applicability of these formulas in numerical calculations is illustrated.     

A note on biorthogonal ensembles

We study multiple orthogonal polynomials in the context of biorthogonal ensembles of random matrices. In these ensembles, the eigenvalue probability density function factorizes into a product of two determinants while the eigenvalue correlation functions can be written as a determinant of a kernel function. We show that the kernel is itself an average of a single ratio of characteristic polynomials. In the same vein, we prove that the type I multiple polynomials can be expressed as an average of the inverse of a characteristic polynomial. We finally introduce a new biorthogonal matrix ensemble, namely the chiral unitary perturbed by a source term, whose multiple polynomials are related to the modified Bessel function of the first kind.     

Wahrscheinlichkeitsfunktionen diskreter Verteilungen als Lösungen der Pearsonschen Differenzengleichung für die diskreten klassischen Orthogonalpolynome

Using the Pearson difference equation for the discrete classical orthogonal polynomials the difference equations and the Rodrigues formulas are obtained. The resulting weight functions prove to be the probability functions of the most important discrete probability distributions: Pólya distribution from the Hahn and Krawtchouk polynomials, negative binomial distribution from the Meixner polynomials, Poisson distribution from the Charlier polynomials.     

Approximating probability density functions and their convolutions using orthogonal polynomials

This paper describes and tests methods for piecewise polynomial approximation of probability density functions using orthogonal polynomials. Empirical tests indicate that the procedure described in this paper can provide very accurate estimates of probabilities and means when the probability density function cannot be integrated in closed form. Furthermore, the procedure lends itself to approximating convolutions of probability densities. Such approximations are useful in project management, inventory modeling, and reliability calculations, to name a few applications. In these applications, decision makers desire an approximation method that is robust rather than customized. Also, for these applications the most appropriate criterion for accuracy is the average percent error over the support of the density function as opposed to the conventional average absolute error or average squared error. In this paper, we develop methods for using five well-known orthogonal polynomials for approximating density functions and recommend one of them as giving the best performance overall.     

Derivatives of a finite class of orthogonal polynomials related to inverse Gamma distribution

In this work, we consider derivatives of a finite class of orthogonal polynomials with respect to weight function which is related to the probability density function of the inverse gamma distribution over the positive real line. General properties for this derivative class such as orthogonality, Rodrigues’ formula, recurrence relation, generating function and various other related properties such as self-adjoint form and normal form are indicated. The corresponding Gaussian quadrature formulae are introduced with examples. These examples are provided to support the advantages of considering the derivatives class of the finite class of orthogonal polynomials related to inverse gamma distribution. The orthogonality property related to the Fourier transform of the derivative class under discussion is also given.     

Appell pseudopolynomials and Erlang-type risk models

Appell polynomials are known to play a key role in certain first-crossing problems. The present paper considers a rather general insurance risk model where the claim interarrival times are independent and exponentially distributed with different parameters, the successive claim amounts may be dependent and the premium income is an arbitrary deterministic function. It is shown that the non-ruin (or survival) probability over a finite horizon may be expressed in terms of a remarkable family of functions, named pseudopolynomials, that generalize the classical Appell polynomials. The presence of that underlying algebraic structure is exploited to provide a closed formula, almost explicit, for the non-ruin probability.     

On the Solution of High Dimensional Fokker Planck Equations using Orthogonal Polynomial Expansion

Solving the Fokker-Planck-Equation for multidimensional nonlinear systems is a great challenge in the field of stochastic dynamics. As for many mechanical systems a general idea about the shape of stationary solutions for the probability density function is known, it seems promising to use an approach that contains this knowledge. This is done using a Galerkin-method which applies approximate solutions as weighting functions for the expansion of orthogonal polynomials, e.g. generalized Hermite polynomials [1]. As examples, nonlinear oscillators containing cubical restoring (Duffing oscillators) and cubical damping elements are considered. The method is applied to the two-dimensional problem of a single-degree-of-freedom oscillator and consecutively extended up to dimension ten. Results for probability density functions are presented and compared with results from Monte Carlo simulations. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Romanovski–Jacobi polynomials and their related approximation results

The aim of this article is to present the essential properties of a finite class of orthogonal polynomials related to the probability density function of the F -distribution over the positive real line. We introduce some basic properties of the Romanovski–Jacobi polynomials, the Romanovski–Jacobi–Gauss type quadrature formulae and the associated interpolation, discrete transforms, spectral differentiation and integration techniques in the physical and frequency spaces, and basic approximation results for the weighted projection operator in the nonuniformly weighted Sobolev space. We discuss the relationship between such kinds of finite orthogonal polynomials and other classes of infinite orthogonal polynomials. Moreover, we derive spectral Galerkin schemes based on a Romanovski–Jacobi expansion in space and time to solve the Cauchy problem for a scalar linear hyperbolic equation in one and two space dimensions posed in the positive real line. Two numerical examples demonstrate the robustness and accuracy of the schemes.     

Universal Algorithms for Learning Theory. Part II: Piecewise Polynomial Functions

This paper is concerned with estimating the regression function f_ρ in supervised learning by utilizing piecewise polynomial approximations on adaptively generated partitions. The main point of interest is algorithms that with high probability are optimal in terms of the least square error achieved for a given number m of observed data. In a previous paper [1], we have developed for each β > 0 an algorithm for piecewise constant approximation which is proven to provide such optimal order estimates with probability larger than 1- m^-β. In this paper we consider the case of higher-degree polynomials. We show that for general probability measures ρ empirical least squares minimization will not provide optimal error estimates with high probability. We go further in identifying certain conditions on the probability measure ρ which will allow optimal estimates with high probability.     

Remarkable Haar spaces of multivariate piecewise polynomials

Some families of Haar spaces in \(\mathbb {R}^{d},~ d\ge 1,\) whose basis functions are d-variate piecewise polynomials, are highlighted. The starting point is a sequence of univariate piecewise polynomials, called Lobachevsky splines, arised in probability theory and asymptotically related to the normal density function. Then, it is shown that d-variate Lobachevsky splines can be expressed as products of Lobachevsky splines. All these splines have simple analytic expressions and subsets of them are suitable for scattered data interpolation, allowing efficient computation and plain error analysis.     

Numerical computation of characteristic polynomials of Boolean functions and its applications

We study the problem of evaluation of characteristic polynomials of Boolean functions with applications to combinational circuit verification. Two Boolean functions are equivalent if and only if their corresponding characteristic polynomials are identical. However, to verify the equivalence of two Boolean functions it is often impractical to construct the corresponding characteristic polynomials due to a possible exponential blow-up of the terms of the polynomials. Instead, we compare their values at a sample point without explicitly constructing the characteristic polynomials. Specifically, we sample uniformly at random in a unit cube and determine whether two characteristic polynomials are identical by their evaluations at the sample point; the error probability is zero when there are no round-off errors. In the presence of round-off errors, we sample on regular grids and analyze the error probability. We discuss in detail the Shannon expansion for characteristic polynomial evaluation. This revised version was published online in June 2006 with corrections to the Cover Date.     

A stochastic characterization of Hermite polynomials

In this paper, we characterize the Hermite polynomials via stochastic tools: the conditional mean value and the Wiener process. Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, Russia, 1996, Part I.     

Nikol'skij-type and maximal inequalities for generalized trigonometric polynomials

We consider some Nikol'skij-type inequalities, thus inequalities between different metrics of a function, for almost periodic trigonometric polynomials. Some basic methods of probability theory are applied to prove the existence of the distribution function for an almost periodic function in the sense of Besicovitch. Finally, the Maximal function of Hardy and Littlewood is considered and maximal inequalities on Besicovitch spaces are proved. Received: 23 July 1998 / Revised version: 8 March 1999     

Latent class models for time series analysis

Latent class analysis of time series designed to classify and compare sets of series is discussed. For a particular time series in latent class the data are independently normally distributed with a vector of means, and common variance , that is, . The function of time, , can be represented by a linear combination of low-order splines (piecewise polynomials). The probability density function for the data of a time series is posited to be a finite mixture of spherical multivariate normal densities. The maximum-likelihood function is optimized by means of an EM algorithm. The stability of the estimates is investigated using a bootstrap procedure. Examples of real and artificial data are presented. Copyright © 1999 John Wiley & Sons, Ltd.     

On discrete distributions of orderk

Summary The class of discrete distributions of orderk is defined as the class of the generalized discrete distributions with generalizer a discrete distribution truncated at zero and from the right away fromk+1. The probability function and factorial moments of these distributions are expressed in terms of the (right) truncated Bell (partition) polynomials and several special cases are briefly examined. Finally a Poisson process of orderk, leading in particular to the Poisson distribution of orderk, is discussed.     

On a waiting time distribution in a sequence of Bernoulli trials

In the present article we investigate the exact distribution of the waiting time for the r-th non-overlapping appearance of a pair of successes separated by at mosk k–2 failures (k2) in a sequence of independent and identically distributed (iid) Bernoulli trials. Formulae are provided for the probability distribution function, probability generating function and moments and some asymptotic results are discussed. Expressions in terms of certain generalised Fibonacci numbers and polynomials are also included.     

An analogue of the big <Emphasis Type="Italic">q</Emphasis>-Jacobi polynomials in the algebra of symmetric functions

It is well known how to construct a system of symmetric orthogonal polynomials in an arbitrary finite number of variables from an arbitrary system of orthogonal polynomials on the real line. In the special case of the big q-Jacobi polynomials, the number of variables can be made infinite. As a result, in the algebra of symmetric functions, there arises an inhomogeneous basis whose elements are orthogonal with respect to some probability measure. This measure is defined on a certain space of infinite point configurations and hence determines a random point process.     

Probabilistic spherical Marcinkiewicz–Zygmund inequalities

Recently, norm equivalences between spherical polynomials and their sample values at scattered sites have been proved. These so-called Marcinkiewicz–Zygmund inequalities involve a parameter that characterizes the density of the sampling set and they are applicable to all polynomials whose degree does not exceed an upper bound that is determined by the density parameter. We show that if one is satisfied by norm equivalences that hold with prescribed probability only, then the upper bound for the degree of the admissible polynomials can be enlarged significantly and that then, moreover, there exist fixed sampling sets which work for polynomials of all degrees.     

Choquet order for spectra of higher Lamé operators and orthogonal polynomials

We establish a hierarchy of weighted majorization relations for the singularities of generalized Lamé equations and the zeros of their Van Vleck and Heine–Stieltjes polynomials as well as for multiparameter spectral polynomials of higher Lamé operators. These relations translate into natural dilation and subordination properties in the Choquet order for certain probability measures associated with the aforementioned polynomials. As a consequence we obtain new inequalities for the moments and logarithmic potentials of the corresponding root-counting measures and their weak-^* limits in the semi-classical and various thermodynamic asymptotic regimes. We also prove analogous results for systems of orthogonal polynomials such as Jacobi polynomials.     

Numerical Methods for Approximating Continuous Probability Density Functions, over [0, {infty}), Using Moments

Three numerical methods are presented for the reconstructionof a continuous probability density function f(x) from givenvalues of the moments of the distribution. The first methodis obtained by assuming that f(x) may be expanded as an infiniteseries of generalized Laguerre polynomials . The use of ordinary Laguerre polynomials, corresponding to theparticular choice = 0, is related to a second method involvingthe numerical inversion of a Laplace transform. In the thirdmethod the principle of maximization of entropy, subject tothe known moment constraints, is used to reconstruct f(x). Thetype of fit to be expected from each method is illustrated bynumerical examples.