# Consistent Prediction Of Stationary Ergodic Sequences

We are interested in stationary ergodic sequences . A striking result by Bailey says that there does not exist a predictor of such that

holds for any stationary ergodic sequence . (In other words, there does no exist a universal strongly consistent predictor.) This is true even for binary sequences .

Examples for binary sequences were constructed independently by David Bailey (1976) and, later, by Boris Ryabko (1988). Ryabko's example is much simpler; his intuitive exposition was interpreted and extended in Gyorfi et al. (1998), too.

A predictor of is called *universally consistent* if

holds for any bounded stationary ergodic sequence . The existence of a universally consistent predictor was established by Morvai et al. (1997).

### Bibliography

- D. H. Bailey (1976). Sequential schemes for classifying and predicting ergodic processes. PhD dissertation, Stanford University, Stanford, CA.
- L. Gyorfi, G. Morvai, and S. J. Yakowitz (1998). Limits to consistent on-line forecasting for ergodic time series.
*IEEE Transactions on Information Theory*44:886 - 892. - G. Morvai, S. J. Yakowitz, and P. Algoet (1997). Weakly convergent non-parametric forecasting of stationary time series.
*IEEE Transactions on Information Theory*43:483 - 498. - B. Ryabko (1998). Prediction of random sequences and universal coding.
*Problems of Information Transmission*24:87 - 96.