Stable mixed moving averages

Summary The class of (non-Gaussian) stable moving average processes is extended by introducing an appropriate joint randomization of the filter function and of the stable noise, leading to stable mixed moving averages. Their distribution determines a certain combination of the filter function and the mixing measure, leading to a generalization of a theorem of Kanter (1973) for usual moving averages. Stable mixed moving averages contain sums of independent stable moving averages, are ergodic and are not harmonizable. Also a class of stable mixed moving averages is constructed with the reflection positivity property.Research supported by AFSOR Contract 91-0030Research also supported by ARO DAAL-91-G-0176Research also supported by AFOSR 90-0168Research also supported by ONR N00014-91-J-0277     

Local ergodic theorems for K-spherical averages on the Heisenberg group

Given a Gelfand pair where is the Heisenberg group and K is a compact subgroup of the unitary group U(n) we consider the sphere and ball averages of certain K-invariant measures on . We prove local ergodic theorems for these measures when . We also consider averages over annuli in the case of reduced Heisenberg group and show that when the functions have zero mean value the maximal function associated to the annulus averages behave better than the spherical maximal function. We use square function arguments which require several properties of the K-spherical functions. Received September 1, 1998; in final form July 9, 1999     

On the rate of convergence in the individual ergodic theorem for the action of a semigroup

We consider the individual ergodic theorem for the action of a semigroup of measurepreserving mappings. We estimate the rate of convergence using estimates for the probability of large deviations for the ergodic averages with an essentially bounded averaging function. We find estimates for the rate of convergence of the ergodic averages in the cases of Benedicks–Carleson quadratic mappings, expanding mappings of Pomeau–Manneville type with a neutral point, and multidimensional shifts.     

From integral estimates of functions to uniform and locally averaged ones

Obtaining pointwise estimates from above for a function or its averages, under known integral restrictions on the growth of the function, often arise in the function theory. We offer an approach to such problems based on integral Jensen’s inequality with a convex function.     

Some a priori estimates of solutions to the Maxwell equations

In this paper, we present a collection of a priori estimates of the electromagnetic field scattered by a general bounded domain. The constitutive relations of the scatterer are in general anisotropic. Surface averages are investigated, and several results on the decay of these averages are presented. The norm of the exterior Calderón operator for a sphere is investigated and depicted as a function of the frequency. Copyright © 2014 John Wiley & Sons, Ltd.     

Limit theorems for additive statistics based on moving average samples

We study statistics based on samples of moving averages generated by stationary sequence of random variables. The central limit theorem (CLT) is proved for sequences of observations defined by an analytic function of moving averages under consideration. For U- and V -statistics with canonical (degenerate) kernels, the limit distributions are studied.     

Oscillation in ergodic theory: Higher dimensional results

In this paper we continue our investigations of square function inequalities. The results in [9] are primarily one dimensional, and here we extend all the results to multi-dimensional averages. Our basic tool is still a comparison of the ergodic averages with various dyadic (reversed) martingales, but the Fourier transform arguments are replaced by more geometric almost orthogonality arguments. The results imply the pointwise ergodic theorem for the action of commuting measure preserving transformations, and give additional information such as control of the number of upcrossings of the ergodic averages. Related differentiation results are also discussed. R. Jones is partially supported by NSF Grant DMS-9531526. J. Rosenblatt is partially supported by NSF Grant DMS-9705228. M. Wierdl is partially supported by NSF Grant DMS-9801602.     

Applications of the L-functions ratios conjectures

In upcoming papers by Conrey, Farmer and Zirnbauer there appearconjectural formulas for averages, over a family, of ratiosof products of shifted L-functions. In this paper we will presentvarious applications of these ratios conjectures to a wide varietyof problems that are of interest in number theory, such as lowerorder terms in the zero statistics of L-functions, mollifiedmoments of L-functions and discrete averages over zeros of theRiemann zeta function. In particular, using the ratios conjectureswe easily derive the answers to a number of notoriously difficultcomputations.     

Average sampling in spline subspaces

We show that every function in a spline subspace is uniquely determined and can be reconstructed by its local averages near certain points. Regular and irregular average sampling theorems for spline subspaces are obtained.     

Inversion of the local Pompeiu transformation on Riemannian symmetric spaces of rank one

We study the problem of inversion of the local Pompeiu transform on Riemannian symmetric spaces of rank one. The explicit formula for the reconstruction of a function by its averages over balls and spheres with a single fixed radius is obtained.     

A general cooperation theorem for hypercycles

We derive a condition for a closed invariant subset of a compact dynamical system to be an attractor (resp. repellor) combining the usual Ljapunov function methods with time averages. Applications are given to concrete systems endowed with some cyclic symmetry. In particular, cooperation of the inhomogeneous hypercycle is shown.     

Random algorithms for convex minimization problems

This paper deals with iterative gradient and subgradient methods with random feasibility steps for solving constrained convex minimization problems, where the constraint set is specified as the intersection of possibly infinitely many constraint sets. Each constraint set is assumed to be given as a level set of a convex but not necessarily differentiable function. The proposed algorithms are applicable to the situation where the whole constraint set of the problem is not known in advance, but it is rather learned in time through observations. Also, the algorithms are of interest for constrained optimization problems where the constraints are known but the number of constraints is either large or not finite. We analyze the proposed algorithm for the case when the objective function is differentiable with Lipschitz gradients and the case when the objective function is not necessarily differentiable. The behavior of the algorithm is investigated both for diminishing and non-diminishing stepsize values. The almost sure convergence to an optimal solution is established for diminishing stepsize. For non-diminishing stepsize, the error bounds are established for the expected distances of the weighted averages of the iterates from the constraint set, as well as for the expected sub-optimality of the function values along the weighted averages.     

A variant of the theory of local strains

A variant of the theory of local strains giving the stress tensor as a function of a given strain tensor is formulated. The stresses in an orthogonal coordinate system are established by means of a functional that averages the local stresses expressed by the local stress function. This function is determined by the given strain program. It is shown that in certain practical problems these relations are more convenient than those previously proposed.Mekhanika Polimerov, Vol. 3, No. 5, pp. 800–802, 1967     

Average conditions for Kolmogorov systems

In this paper we consider nonautonomous competitive Kolmogorov systems, which is a generalization of classical Lotka–Volterra competition model. Applying Ahmad and Lazer’s definitions of lower and upper averages of a function, we give the average conditions for the persistence, permanence and global attractivity of the system.     

两类平均及其应用

本文给出了两类平均,一类是单指数函数的a次幂平均,这类平均仅与函数的端点值有关;另一类平均是倒数平均。还给出了这两类平均在医学应用中的实例。     

Using composite moving averages to forecast sales

Combining moving averages has been suggested as a simple and practical means to improve sales forecasting. Here we present a natural extension whereby combinations of all possible moving averages up to a given number of periods are employed. We evaluate the method's performance relative to other methods, such as simple moving averages and exponentially-weighted moving averages, on two industrial data sets. Particular attention is placed on methods for selecting the number of periods employed, and on handling noisy data.     

Oracle inequalities for support vector machines that are based on random entropy numbers

In this paper, we present a new technique for bounding local Rademacher averages of function classes induced by a loss function and a reproducing kernel Hilbert space (RKHS). At the heart of this technique lies the observation that certain expectations of random entropy numbers can be bounded by the eigenvalues of the integral operator associated with the RKHS. We then work out the details of the new technique by establishing two new oracle inequalities for support vector machines, which complement and generalize previous results.     

Analysis to Neyman-Pearson classification with convex loss function

Neyman-Pearson classification has been studied in several articles before.But they all proceeded in the classes of indicator functions with indicator function as the loss function,which make the calculation to be difficult.This paper investigates NeymanPearson classification with convex loss function in the arbitrary class of real measurable functions.A general condition is given under which Neyman-Pearson classification with convex loss function has the same classifier as that with indicator loss function.We give analysis to NP-ERM with convex loss function and prove it's performance guarantees.An example of complexity penalty pair about convex loss function risk in terms of Rademacher averages is studied,which produces a tight PAC bound of the NP-ERM with convex loss function.     

A Unified Approach to Several Inequalities Involving Functions and Derivatives

There are many inequalities measuring the deviation of the average of a function over an interval from a linear combination of values of the function and some of its derivatives. A general setting is given from which the desired inequalities are obtained using Holder's inequality. Moreover, sharpness of the constants is usually easy to prove by studying the equality cases of Holder's inequality. Comparison of averages, extension to weighted integrals and n-dimensional results are also given.     

Weak and universal consistency of moving weighted averages

The properties of weighted averages as linear estimators of a regression function and its derivatives are investigated for the fixed design case. Results on weak consistency and on universal consistency are derived, using a modification of the definition of Stone [10]. As examples we consider kernel estimates and weighted local regression estimators and show that the general results apply.