Random difference equations and Renewal theory for products of random matrices

@article{Kesten1973RandomDE,
  title={Random difference equations and Renewal theory for products of random matrices},
  author={Harry Kesten},
  journal={Acta Mathematica},
  year={1973},
  volume={131},
  pages={207-248}
}
  • H. Kesten
  • Published 1 December 1973
  • Mathematics
  • Acta Mathematica
where Mn and Qn are random d • d matrices respectively d-vectors and Yn also is a d-vector. Throughout we take the sequence of pairs (Mn, Q~), n >/1, independently and identically distributed. The equation (1.1) arises in various contexts. We first met a special case in a paper by Solomon, [20] sect. 4, which studies random walks in random environments. Closely related is the fact tha t if Yn(i) is the expected number of particles of type i in the nth generation of a d-type branching process in… 
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TLDR
This work characterize the distribution of the first passage time of the matrix recursive sequence V_n = M_n V_{n-1} + Q_n, showing that this distribution converges to the stationary law of the exponentially-shifted Markov random walk and describes the large exceedance paths via two conditioned limit laws.
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