Raymond Cheng

Weak parallelogram laws on banach spaces and applications to prediction

This paper concerns a family of weak parallelogram laws for Banach spaces. It is shown that the familiar Lebesgue spaces satisfy a range of these inequalities. Connections are made to basic geometric ideas, such as smoothness, convexity, and Pythagorean-type theorems. The results are applied to the linear prediction of random processes spanning a Banach space. In particular, the weak parallelogram laws furnish coefficient growth estimates, Baxter-type inequalities, and criteria for regularity.

Regularity and Minimality of Infinite Variance Processes

For stationary processes with infinite variance the notions of covariance and spectrum are not defined. We characterize regularity and minimality of such processes in the spirit of some classical results for second-order processes, namely values of the process forming conditional basis for their spans. Several open problems are discussed.

Prediction with incomplete past and interpolation of missing values

A generalized innovation algorithm is used to solve the problems of prediction of future values based on incomplete past and interpolation of missing values of a stationary time series. The emphasis is on the computational aspects and the proposed method is particularly useful when there are several missing values with arbitrary patterns.

The mixing rate of a stationary multivariate process

It is known that for a broad class of multivariate stationary processes, the degree of smoothness of the spectral density function reveals the rates of various types of strong mixing. Issues related to the spectral factorization of the density function play a central role.

Ho¨lder classes of vector-valued functions and convergence of the best predictor

The Holder classes [Lambda]a of vector-valued functions are defined. The functions in each space [Lambda]a are completely characterized by conditions concerning the decay of their Fourier coefficients, their smoothness, and their approximability by polynomials. It is shown that, in some sense, [Lambda]a is closed under multiplication, inversion, and factorization. These ideas are applied to a prediction problem for multivariate stationary processes. Specifically, spectral criteria are derived for the convergence rate of the series representation for the best linear predictor.

Mate choice and the evolutionary stability of a fixed threshold in a sequential search strategy

The sequential search strategy is a prominent model of searcher behavior, derived as a rule by which females might sample and choose a mate from a distribution of prospective partners. The strategy involves a threshold criterion against which prospective mates are evaluated. The optimal threshold depends on the attributes of prospective mates, which are likely to vary across generations or within the lifetime of searchers due to stochastic environmental events. The extent of this variability and the cost to acquire information on the distribution of the quality of prospective mates determine whether a learned or environmentally canalized threshold is likely to be favored. In this paper, we determine conditions on cross-generational perturbations of the distribution of male phenotypes that allow for the evolutionary stability of an environmentally canalized threshold. In particular, we derive conditions under which there is a genetically determined threshold that is optimal over an evolutionary time scale in comparison to any other unlearned threshold. These considerations also reveal a simple algorithm by which the threshold could be learned.

Sequential Search Beats Best-of-N Search

Abstract It is proved that the optimized sequential search strategy is superior to the best-of-N search strategy, in terms of expected net return, for all possible distributions and utility functions, and under a fixed cost per prospect.

Multipliers of Sequence Spaces

Abstract This paper is selective survey on the space lAp and its multipliers. It also includes some connections of multipliers to Birkhoff-James orthogonality

Sequential Search Beats a Two-Parameter Search

Abstract A search strategy is introduced that combines features of both the sequential and best-of-N search strategies. This hybrid search strategy has two parameters, an acceptance threshold T and a maximum search length N. It is shown that the optimized sequential search strategy nonetheless dominates the hybrid strategy for all values of T and N and for all underlying distributions, assuming a fixed cost per prospect.

Rational spectral densities and strong mixing

This article is concerned with stationary random fields with rational spectral densities. The dichotomy between the one-parameter and multiparameter cases is explored, particularly in terms of a strong mixing condition. The rich variety of behavior exhibited by the multiparameter case is demonstrated.