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

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

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

Spectral density estimation for stationary stable processes

Weakly and strongly consistent nonparametric estimates, along with rates of convergence, are established for the spectral density of certain stationary stable processes. This spectral density plays a role, in linear inference problems, analogous to that played by the usual power spectral density of second order stationary processes

Trapezoidal Monte Carlo Integration

The approximation of weighted integrals of random processes by the trapezoidal rule based on an ordered random sample is considered. For processes that are once mean-square continuously differentiable and for weight functions that are twice continuously differentiable, it is shown that the rate of convergence of the mean-square integral approximation error is precisely n-4, and the asymptotic constant is also determined

On the rate distortion function of a memoryless gaussian vector source whose components have fixed variances

Consider a memoryless Gaussian vector source and assume that the variances of the components of its i.i.d. vectors are known and that no further information on their covariance structure is available. We study the dependence of the rate distortion function of such a source on the covariance structure of its i.i.d. vectors. In particular we give a lower bound of the rate distortion function, expressed only in terms of the known variances, and we show that it is tighter than the Shannon lower bound over a certain distortion region

Ergodic properties of stationary stable processes

We derive spectral necessary and sufficient conditions for stationary symmetric stable processes to be metrically transitive and mixing. We then consider some important classes of stationary stable processes: Sub-Gaussian stationary processes and stationary stable processes with a harmonic spectral representation are never metrically transitive, the latter in sharp contrast with the Gaussian case. Stable processes with a harmonic spectral representation satisfy a strong law of large numbers even though they are not generally stationary. For doubly stationary stable processes, sufficient conditions are derived for metric transitivity and mixing, and necessary and sufficient conditions for a strong law of large numbers

On the Conditional Variance for Scale Mixtures of Normal Distributions

For a scale mixture of normal vector, X = A1/2G, where X G ∈ Rnand A is a positive variable, independent of the normal vector G, we obtain that the conditional variance covariance, Cov(X2 X1), is always finite a,s for m ≥ 2, where X1∈ Rnand m < n, and remains a.s. finite even for m = 1, if and only if the square root moment of the scale factor is finite. It is shown that the variance is not degenerate as in the Gaussian case, but depends upon a function SA,m(.) for which various properties are derived. Application to a uniform and stable scale of normal distributions are also give

Characterization of linear and harmonizable fractional stable motions

AbstractWe characterize the linear and harmonizable fractional stable motions as the self-similar stable processes with stationary increments whose left-equivalent (or right-equivalent) stationary processes are moving averages and harmonizable respectively