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We proposed a learning algorithm for nonparametric estimation and on-lineprediction for general stationary ergodic sources. We prepare histograms eachof which estimates the probability as a finite distribution, and mixture themwith weights to construct an estimator. The whole analysis is based on measuretheory. The estimator works whether the source is discrete or continuous. If itis stationary ergodic, then the measure theoretically given Kullback-Leiblerinformation divided by the sequence length $n$ converges to zero as $n$ goes toinfinity. In particular, for continuous sources, the method does not requireexistence of a probability density function.

Nonparametric Estimation and On-Line Prediction for General Stationary Ergodic Sources