Vol. 45, No. 2 DUKE MATHEMATICAL JOURNAL(C) June 1978 A LIMIT THEOREM FOR MARKOV CHAINS IN COMPACT METRIC SPACES WITH APPLICATIONS TO PRODUCTS OF RANDOM MATRICES THOMAS KAIJSER CONTENTS 1. Introduction 312 1.1. Condition C 312 1.2. A special type of Markov chains 312 1.3. Theorem 312 1.4. The connection with products of random matrices 314 1.5. A historical remark 314 1.6. A result on products of random matrices with negative elements 315 PART I THE PROOF OF THEOREM 2. Preliminaries 316 3. An e-net 320 4. The main lemma 321 5. The final step 324 6. Some comments 326 PART II ON PRODUCTS OF RANDOM MATRICES 7. Preliminaries 327 8. On products of positive matrices 329 9. On orthogonal matrices 330 10. On 2 2 matrices 331 11. A simple case 332 12. A more difficult case 335 13. The main estimate 338 14. An upper bound 342 15. ConditionG’ holds 344 16. A generalization 346 Received August 12, 1977. Revision received February 6, 1978. 311