## Sequential schemes for classifying and predicting ergodic processes |

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### Contents

Determination of Entropy | 6 |

Classification of Processes | 17 |

Determination of Conditional Probabilities | 27 |

Copyright | |

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1-block actual entropy arbitrary order Bernoulli shift classes of processes conditional probability denote the number denote the string diagram may assist digit in position digits printed Doctor of Philosophy dominated convergence theorem el,l ergodic theorem existence Fatou's lemma finitely often a.e. gambler given the infinite Given w e gn(w i=l i=l implies w e infinite past infinitely many different Information Theory initial segment integers iterated logarithm lemma log pjw M-order mixing Markov m-strings mixing Markov process notation Note number of occurrences O's and l's partition set Pk(w problem process is independent processes of arbitrary qk qk qk qk+l qk t qk qk-l random variable required property result sequential scheme set of sequences set Q stack of height substring Suppose Theorem 1.1 Ti+1w Tn(w universal gambling scheme unknown ergodic process unknown process vn(w wager wffi X^Cw XQ(w