1. T.M. Adams and A.B. Nobel, "On density estimation from an ergodic process", Annals of Probability, vol. 26, pp.794-804, 1998.

Summary: This paper answers an open question concerning the existence of universal density estimation procedures for stationary ergodic processes. It is shown that no procedure can consistently estimate the one-dimensional marginal density of every stationary ergodic process for which such a density exists. Given any candidate density estimation scheme, a cutting and stacking argument is used to construct an ergodic process whose marginal distribution is uniform on [0,1], but such that estimates produced by the given scheme fail to converge.

2. A.B. Nobel, "Limits to classification and regression estimation from ergodic process", Annals of Statistics, vol. 27, pp.262-273, 1999.

Summary: The paper answers two open questions concerning the existence of universal procedures for classification and regression estimation from stationary ergodic processes. It is shown that no procedure can produce weakly consistent regression estimates from every bivariate stationary ergodic process, even if the covariate and response variables are restricted to take values in the unit interval. It is further shown that no procedure can produce weakly consistent classification rules from every bivariate stationary ergodic process. The results of the paper are derived via reduction arguments, and are based in part on the results of Adams and Nobel (1998).

3. A.B. Nobel, "Consistent estimation of a dynamical map", Nonlinear Dynamics and Statistics, A.I. Mees editor, Birkhauser, 2001.

Summary: This paper considers estimation of a non-linear map F from R^d to itself that governs the evolution of an observed dynamical system. Two dynamical models are studied. In the first model, F is successively applied to a fixed initial vector in the absence of noise, so that the observed states of the system constitute a trajectory of F. In the second, dynamical noise model, the system is perturbed by independent noise between each application of F. Estimates of F are proposed for each model, and are shown to be consistent under general conditions. Throughout, no assumptions are made regarding mixing rates of the observations. Both continuous and more general measurable maps F are considered.

4. T.M. Adams and A.B. Nobel, "Finitary reconstruction of a measure preserving transformation", Israel Journal of Mathematics, vol.126, pp.309-326, 2001.

Summary: This paper considers the finitary reconstruction of a measure preserving transformation of a complete separable metric space. Observations may come from a single trajectory or, more generally, a suitable reconstruction sequence. A finitary estimation scheme is described and shown to be consistent in the weak topology whenever the asymptotic distribution of the observations is comparable to a known reference measure. In general, the target transformation need not be ergodic. It is also shown that no finitary estimation scheme is consistent in the strong topology for the family of all ergodic Lebesgue measure preserving transformations of the unit interval.

5. A.B. Nobel and T.M. Adams, "On regression estimation from ergodic samples with additive noise", IEEE Transactions on Information Theory, vol.47, pp.2895-2902, 2001.

Summary: We study the problem of estimating an unknown function from ergodic samples corrupted by additive noise. It is shown that one can consistently recover an unknown measurable function in this setting if the one dimensional distribution of the samples is comparable to a known reference distribution, and the noise is independent of the samples and has known mixing rates. The estimates are applied to deterministic sampling schemes, in which successive samples are obtained by repeatedly applying a fixed map to a given initial vector, and it is then shown how the estimates can be used to reconstruct an ergodic transformation from one of its trajectories.

6. A.B. Nobel, "On optimal sequential prediction schemes for general processes", Submitted for publication.

Summary: The subject of this paper is the stochastic sequential prediction problem. The first part of the paper is devoted to some basic properties of Cesaro and strongly optimal decision schemes for general processes and strictly convex loss functions. It is shown in each case that optimal schemes are unique in a natural sense, and that optimality is equivalent to a form of calibration. For binary processes it is shown that thresholding an optimal prediction scheme for the squared loss yields an optimal binary prediction scheme for the Hamming loss. In the second part of the paper some results of Algoet on the existence of Cesaro optimal schemes for the family ergodic processes are rederived in a direct way using aggregating methods originally developed for individual sequences. The properties of such schemes are briefly considered.