This article gives sufficient conditions for the limit distribution of products of i.i.d. 2 × 2 stochastic matrices to be continuous singular, when the support of the distribution of the individual random matrices is countably infinite. It extends a previous result for which the support of the random matrices is finite. The result is based on adapting existing proofs in the context of attractors and iterated function systems to the case of infinite iterated function systems.
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Dubins L E and Freedman D A, Invariant probabilities for certain Markov processes, Ann. Math. Stat. 37 (1966) 837–847
Edgar G A, Integral, probability, and fractal measures (New York: Springer-Verlag) (1998)
Edgar G A, Measure, topology, and fractal geometry, Undergraduate Texts in Mathematics (New York: Springer-Verlag) (1990)
Erdös P, On a family of symmetric Bernoulli convolutions, Am. J. Math. 61 (1939) 974–976
Erdös P, On the smoothness properties of Bernoulli convolutions, Am. J. Math. 62 (1940) 180–186
Mukherjea A, Topics in products of random matrices, Tata Institute of Fundamental Research Lectures on Mathematics 87 (Mumbai) (2000)
Mukherjea A, Nakassis A and Ratti J, On the distribution of the limit of products of i.i.d. 2 by 2 random stochastic matrices, J. Theoret. Probab. 12 (1999) 571–583
Mukherjea A and Tserpes N A, Measures on topological semigroups, Lecture Notes in Math. 547 (Springer) (1976)
Peres Y and Solomyak B, Absolute continuity of Bernoulli convolutions, a simple proof, Math. Res. Lett. 3 (1996)
Rosenblatt, Murray, Markov Processes: Structure and Asymptotic Behavior (Springer Verlag) (1971)
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Mukherjea, A., Restrepo, R. Upper packing dimension of a measure and the limit distribution of products of i.i.d. stochastic matrices. Proc Math Sci 119, 669 (2009). https://doi.org/10.1007/s12044-009-0052-x
- Packing dimension
- stochastic matrices
- Erdös sum
- products of random matrices
- continuous singularity of the limit distribution