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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References

SHOWING 1-10 OF 27 REFERENCES
Limit laws of a sequence determined by a random difference equation governing a one-compartment system
Probability measures on metric spaces
Markov Processes, Structure and Asymptotic Behavior
I Basic Notions and Illustrations.- 0. Summary.- 1. Markov Processes and Transition Probability Functions.- 2. Markov Chains.- 3. Independent Random Variables.- 4. Some Continuous Parameter Markov…
Models for cultural inheritance. I. Group mean and within group variation.
Probability measures on metric spaces
The Borel subsets of a metric space Probability measures in a metric space Probability measures in a metric group Probability measures in locally compact abelian groups The Kolmogorov consistency…
Stability of a linear difference system with random parameters
Necessary and sufficient conditions are derived for the mean-square exponential stability of a system described by an n-th order stochastic difference equation.
An Introduction to Probability Theory and Its Applications
  • 1967
A n introduction to probability theory and it~
  • applications, Vol. I I ,
  • 1971
A survey on the theory of random equations
Additional Limit Theorems for Indecomposable Multidimensional Galton-Watson Processes
...
...