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Generalized $\raisebox{0.1ex}{$\phi$ }$-Divergence
and
Frequency Analysis in Markov Chains





Dissertation


zur Erlangung des Doktorgrades
an der Naturwissenschaftlichen Fakultät
der Universität Salzburg
eingereicht von


Stefan Wegenkittl






4. Mai 1998

No one should be expected
to solve a math problem
that has a ``twelve'' in it.


- Peppermint Patty, Peanuts







It is difficult to predict,
especially the future.


- N. Bohr, old Danish proverb














TO DORIS.


IT WAS A PLEASURE AND A PIECE OF GOOD FORTUNE FOR ME TO BE ABLE TO WRITE THIS THESIS WHILST BEING MEMBER OF THE PLAB GROUP AT SALZBURG.
THANK YOU, PETER, FOR INTRODUCING ME TO THE WORLD OF PSEUDORANDOMNESS AND FOR MAKING ALL THE COLLABORATIONS POSSIBLE. THANK YOU, CHARLY, HANNES, AND KARIN, FOR MAKING MATHEMATICS SO ADVENTUROUS AND COLOURFUL, AND - OF COURSE - FOR THE GOOD MOOD AND MUSIC IN OUR OFFICE.
THE DEVELOPMENT OF THIS WORK WAS ACCOMPANIED BY PROF. F. ÖSTERREICHER FROM THE VERY BEGINNING. THANK YOU, FERDINAND, FOR THE EXCELLENT TEACHING AND YOUR PATIENCE. THE MATERIAL AND PRESENTATION OWES MUCH TO YOUR CONTRIBUTIONS!
THANK YOU, MAX, FOR INTRODUCING ME TO THE BEAUTY OF PROBABILITY.
I GRATEFULLY ACKNOWLEDGE THE ADVICE, COMMENTS, CORRECTIONS AND ENCOURAGEMENTS OF PROF. PIERRE L'ECUYER (WHO WAS THE FIRST TO DRAW MY ATTENTION TO OVERLAPPING POWER DIVERGENCES), PROF. MAKOTO MATSUMOTO (WHO INVENTED THE PRIMARY VERSION OF THE GAMBLING TEST), DR. GRODZICKI (WHO EXPLAINED BOHR'S MODEL OF THE HYDROGEN ATOM TO ME), AND MY SISTER SUSANNE ``SU'' (WHO RENDERED THE SALZBURGIAN ACCENT INTO SOMETHING PRETTY MUCH LIKE BRITISH ENGLISH).

THIS THESIS WOULD NOT HAVE BEEN STARTED OUT WITHOUT THE GENEROUS SUPPORT AND UNDERSTANDNING OF MY PARENTS RENATE AND WILLIBALD, AND MY BROTHER HELMUT AND SISTER SUSANNE.
IT WOULD NEVER HAVE BEEN FINISHED WITHOUT DORIS ALWAYS DRAWING MY ATTENTION TO THE FUNNY SIDE OF EVERYTHING OFTEN TAKEN SO SERIOUSLY. I DEDICATE THIS WORK TO YOU.
SALZBURG, MAI 4, 1998



THIS WORK WAS SUPPORTED BY THE AUSTRIAN SCIENCE FOUNDATION (FWF), PROJECT NO. P11143-MAT LED BY PETER HELLEKALEK.



 
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Stefan Wegenkittl
1998-05-19