Limits to consistent on-line forecasting for ergodic time series

This study concerns problems of time-series forecasting under the weakest of assumptions. Related results are surveyed and are points of departure for the developments here, some of which are new and others are new derivations of previous findings. The contributions in this study are all negative, showing that various plausible prediction problems are unsolvable, or in other cases, are not solvable by predictors which are known to be consistent when mixing conditions hold

Nonparametric sequential prediction for stationary processes

We study the problem of finding an universal estimation scheme $h_n:\mathbb{R}^n\to \mathbb{R}$, $n=1,2,...$ which will satisfy \lim_{t\rightarrow\infty}{\frac{1}{t}}\sum_{i=1}^t|h_ i(X_0,X_1,...,X_{i-1})-E(X_i|X_0,X_1,...,X_{i-1})|^p=0 a.s. for all real valued stationary and ergodic processes that are in $L^p$. We will construct a single such scheme for all $1<p\le\infty$, and show that for $p=1$ mere integrability does not suffice but $L\log^+L$ does.Comment: Published in at http://dx.doi.org/10.1214/10-AOP576 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Weakly Convergent Nonparametric Forecasting of Stationary Time Series

The conditional distribution of the next outcome given the infinite past of a stationary process can be inferred from finite but growing segments of the past. Several schemes are known for constructing pointwise consistent estimates, but they all demand prohibitive amounts of input data. In this paper we consider real-valued time series and construct conditional distribution estimates that make much more efficient use of the input data. The estimates are consistent in a weak sense, and the question whether they are pointwise consistent is still open. For finite-alphabet processes one may rely on a universal data compression scheme like the Lempel-Ziv algorithm to construct conditional probability mass function estimates that are consistent in expected information divergence. Consistency in this strong sense cannot be attained in a universal sense for all stationary processes with values in an infinite alphabet, but weak consistency can. Some applications of the estimates to on-line forecasting, regression and classification are discussed