Random Ergodic Theorems and Markoff Processes with a Stable Distribution
YALE UNIVERSITY New Haven United States
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The purpose of this paper is to discuss the relations between random ergodictheorems and Markoff processes with a stable distribution. Random ergodic theoremconcerning a finite number of measure preserving transformations was obtainedby S. M. Ulam and J. von Neumann. This result was announced in abstractform 6 but the proof has never been published. In the present paper weshall first give a proof of random ergodic theorem concerning a family of infinitelymany measure preserving transformations with a probability distribution on it.We shall then discuss the condition of ergodicity for a family of measure preservingtransformations and its consequence in random ergodic theorems. It turns outthat the theory of Markoff processes with a stable distribution which was previouslydiscussed by J. L. Doob 2, 3, K. Yosida 8, and the author 4 has a veryclose connection with our problem. It will be shown that to any family phi of measurepreserving transformations with a probability distribution there correspondsa Markoff process Ps, B with a stable distribution in such a way that the ergodictheorems concerning the Markoff process Ps, B which were obtained in 8 and4 are nothing but the integrated form of random ergodic theorems concerningthe family phi of measure preserving transformations. Further, the conditions ofergodicity for P correspond exactly to those for phi. It is, indeed, by making use ofthis fact that we prove the equivalence of various conditions of ergodicity for thefamily phi of measure preserving transformations.
- Statistics and Probability