Abstract
Let (X i) be a sequence of m × m i.i.d. stochastic matrices with distribution μ. Then μn is the distribution of X n X n−1 ...X 1. Simple sufficient conditions for the weak convergence of (μn) are presented here. An extremely simple (and verifiable) necessary and sufficient condition is provided for m= 3. The method for m= 3 works for m> 3 even though calculations are more involved for higher values of m. We also discuss the purity of the limit distribution for m≥2.
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Dhar, S., Mukherjea, A. Convergence in Distribution of Products of I.I.D. Nonnegative Matrices. Journal of Theoretical Probability 10, 375–393 (1997). https://doi.org/10.1023/A:1022612516862
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DOI: https://doi.org/10.1023/A:1022612516862