Consistent Prediction Of Stationary Ergodic Gaussian Sequences

Now we are interested in the strongly consistent prediction of stationary ergodic Gaussian sequences $X_1,X_2,\ldots$. Prove or disprove that there exists a predictor $g_n(X_1,\ldots,X_n)$ of $E(X_{n+1}\mid X_1,\ldots,X_n)$ such that

$$
  \lim_{n\to\infty} (g_n(X_1,\ldots,X_n)-E(X_{n+1}\mid X_1,\ldots,X_n))=0
  \quad\text{a.s.}
$$

holds for all stationary ergodic Gaussian sequences $X_1,X_2,\ldots$. (In other words, does there exist a universal strongly consistent predictor for stationary ergodic Gaussian sequences?)

This is known for Cesaro means: there exists a predictor $g_n(X_1,\ldots,X_n)$ of $E(X_{n+1}\mid X_1,\ldots,X_n)$ such that

$$
  \lim_{n\to\infty}
  \frac{1}{n} \sum_{i=1}^n (g_i(X_1,\ldots,X_i)-E(X_{i+1}\mid X_1,\ldots,X_i))^2=0
  \quad\text{a.s.}
$$

holds for all stationary ergodic Gaussian sequences $X_1,X_2,\ldots$. This was proved by Laszlo Gyorfi and Gabor Lugosi (2001).

Under some mild additional assumptions, Dominik Schaefer (2002) demonstrated the existence of a strongly consistent predictor for stationary ergodic Gaussian sequences.

This is also known to fail without the assumption of Gaussianity, even for binary sequences.

This problem was posed by Laszlo Gyorfi et al. (1998).

Bibliography

  • L. Gyorfi and G. Lugosi (2001). Strategies for sequential prediction of stationary time series. In: Modelling Uncertainty: An Examination of its Theory, Methods and Applications, M. Dror, P. L’Ecuyer, F. Szidarovszky (Eds.), pp. 225 - 248, Kluwer.
  • 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.
  • D. Schaefer (2002). Strongly consistent on-line forecasting of centered Gaussian processes. IEEE Transactions on Information Theory 48:791 - 799.