Multiple regression on stable vectors

We give necessary and sufficient conditions for the linearity of multiple regression on a general stable vector, along with a sufficient condition for the finiteness of the conditional absolute moment when 0 < α ≤ 1

Sampling designs for estimation of a random process

A random process X(t), t ∈ [0, 1], is sampled at a finite number of appropriately designed points. On the basis of these observations, we estimate the values of the process at the unsampled points and we measure the performance by an integrated mean square error. We consider the case where the process has a known, or partially or entirely unknown mean, i.e., when it can be modeled as X(t) = m(t)+ N(t), where m(t) is nonrandom and N(t) is random with zero mean and known covariance function. Specifically, we consider (1) the case where m(t) is known, (2) the semiparametric case where m(t) = β1f1(t)+ . + βqfq(t), the βi's are unknown coefficients and the fi's are known regression functions, and (3) the nonparametric case where m(t) is unknown. Here fi(t) and m(t) are of comparable smoothness with the purely random part N(t), and N(t) has no quadratic mean derivative

On the rate distortion functions of memoryless sources under a magnitude-error criterion

We consider the evaluation of and bounds for the rate distortion functions of independent and identically distributed (i.i.d.) sources under a magnitude-error criterion. By refining the ingeneous approach of Tan and Yao we evaluate explicitly the rate distortion functions of larger classes of i.i.d. sources and we obtain families of lower bounds for arbitrary i.i.d. sources

Dyadic sampling approximations for non-sequency-limited signals

Dyadic sampling approximations, as well as error estimates, are derived for non-random signals which are Walsh-Stieltjes transforms and for dyadic-stationary and Walsh-harmonizable random signals. Also derived are inversion formulae for Walsh-Stieltjes transforms, which are used in this paper

Nonorthogonal wavelet approximation with rates of deterministic signals

An nth order asymptotic expansion is produced for the L2-error in a nonorthogonal (in general) wavelet approximation at resolution 2−k of deterministic signals f. These signals over the whole real line are assumed to have n continuous derivatives of bounded variation. The engaged nonorthogonal (in general) scale function ϕ fulfills the partition of unity property, and it is of compact support. The asymptotic expansion involves only even terms of products of integrals involving ϕ with integrals of squares of (the first [ only) derivatives of f

Coherent Orthogonal Polynomials

We discuss as a fundamental characteristic of orthogonal polynomials like the existence of a Lie algebra behind them, can be added to their other relevant aspects. At the basis of the complete framework for orthogonal polynomials we put thus --in addition to differential equations, recurrence relations, Hilbert spaces and square integrable functions-- Lie algebra theory. We start here from the square integrable functions on the open connected subset of the real line whose bases are related to orthogonal polynomials. All these one-dimensional continuous spaces allow, besides the standard uncountable basis ${|x>}$, for an alternative countable basis ${|n>}$. The matrix elements that relate these two bases are essentially the orthogonal polynomials: Hermite polynomials for the line and Laguerre and Legendre polynomials for the half-line and the line interval, respectively. Differential recurrence relations of orthogonal polynomials allow us to realize that they determine a unitary representation of a non-compact Lie algebra, whose second order Casimir ${\cal C}$ gives rise to the second order differential equation that defines the corresponding family of orthogonal polynomials. Thus, the Weyl-Heisenberg algebra $h(1)$ with ${\cal C}=0$ for Hermite polynomials and $su(1,1)$ with ${\cal C}=-1/4$ for Laguerre and Legendre polynomials are obtained. Starting from the orthogonal polynomials the Lie algebra is extended both to the whole space of the ${\cal L}^2$ functions and to the corresponding Universal Enveloping Algebra and transformation group. Generalized coherent states from each vector in the space ${\cal L}^2$ and, in particular, generalized coherent polynomials are thus obtained.Comment: 11 page

A central limit theorem for the sample autocorrelations of a L\'evy driven continuous time moving average process

In this article we consider L\'evy driven continuous time moving average processes observed on a lattice, which are stationary time series. We show asymptotic normality of the sample mean, the sample autocovariances and the sample autocorrelations. A comparison with the classical setting of discrete moving average time series shows that in the last case a correction term should be added to the classical Bartlett formula that yields the asymptotic variance. An application to the asymptotic normality of the estimator of the Hurst exponent of fractional L\'evy processes is also deduced from these results