Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-28T16:51:47.023Z Has data issue: false hasContentIssue false
  • English
  • Français

A Characterisation of Transient Random Walks on Stochastic Matrices with Dirichlet Distributed Limits

Part of: Probability theory on algebraic and topological structures Markov processes Limit theorems

Published online by Cambridge University Press:  19 February 2016

S. McKinlay
S. McKinlay*
Affiliation:
University of Melbourne
*
∗ Postal address: Department of Mathematics and Statistics, University of Melbourne, Parkville 3010, Australia. Email address: s.mckinlay@ms.unimelb.edu.au.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We characterise the class of distributions of random stochastic matrices X with the property that the products X(n)X(n − 1) · · · X(1) of independent and identically distributed copies X(k) of X converge almost surely as n → ∞ and the limit is Dirichlet distributed. This extends a result by Chamayou and Letac (1994) and is illustrated by several examples that are of interest in applications.

Keywords

Products of random matricesDirichlet distributionlimit distributionMarkov chainrandom exchange modelrandom nested simplicesservice networks with polling

MSC classification

Primary: 60J05: Discrete-time Markov processes on general state spaces
Secondary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see ) 60F99: None of the above, but in this section 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 2 , June 2014 , pp. 542 - 555
Copyright
© Applied Probability Trust 

References

Aitchison, J. (1986). The Statistical Analysis of Compositional Data. Chapman and Hall, London.Google Scholar
Bellman, R. (1954). Limit theorems for non-commutative operators. I. Duke Math. J. 21, 491–500.Google Scholar
Bougerol, P. and Lacroix, J. (1985). Products of Random Matrices with Applications to Schrödinger Operators. Birkhäuser, Boston, MA.Google Scholar
Bruneau, L., Joye, A. and Merkli, M. (2010). Infinite products of random matrices and repeated interaction dynamics. Ann. Inst. H. Poincaré Prob. Statist. 46, 442–464.CrossRefGoogle Scholar
Chamayou, J.-F. and Letac, G. (1991). Explicit stationary distributions for compositions of random functions and products of random matrices. J. Theoret. Prob. 4, 3–36.CrossRefGoogle Scholar
Chamayou, J.-F. and Letac, G. (1994). A transient random walk on stochastic matrices with Dirichlet distributions. Ann. Prob. 22, 424–430.Google Scholar
DeGroot, M. H. and Rao, M. M. (1963). Stochastic give-and-take. J. Math. Anal. Appl. 7, 489–498.CrossRefGoogle Scholar
Dufresne, D. (1998). Algebraic properties of beta and gamma distributions, and applications. Adv. Appl. Math. 20, 285–299.CrossRefGoogle Scholar
Furstenberg, H. and Kesten, H. (1960). Products of random matrices. Ann. Math. Statist. 31, 457–469.Google Scholar
Högnäs, G. and Mukherjea, A. (2011). Probability Measures on Semigroups, 2nd edn. Springer, New York.Google Scholar
Holley, R. and Liggett, T. M. (1981). Generalized potlatch and smoothing processes. Z. Wahrscheinlichkeitsth. 55, 165–196.Google Scholar
Kesten, H. and Spitzer, F. (1984). Convergence in distribution of products of random matrices. Z. Wahrscheinlichkeitsth. 67, 363–386.Google Scholar
Letac, G. (2002). Donkey walk and Dirichlet distributions. Statist. Prob. Lett. 57, 17–22.CrossRefGoogle Scholar
Letac, G. and Scarsini, M. (1998). Random nested tetrahedra. Adv. Appl. Prob. 30, 619–627.Google Scholar
Li, J. (1961). Human Genetics. McGraw-Hill, New York.Google Scholar
Liggett, T. M. and Spitzer, F. (1981). Ergodic theorems for coupled random walks and other systems with locally interacting components. Z. Wahrscheinlichkeitsth. 56, 443–468.Google Scholar
Lukacs, E. (1955). A characterisation of the gamma distribution. Ann. Math. Statist. 26, 319–324.Google Scholar
Mukherjea, A. (2000). Topics in Products of Random Matrices. Narosa, New Delhi.Google Scholar
Ng, K. W., Tian, G.-L. and Tang, M.-L. (2011). Dirichlet and Related Distributions: Theory, Methods and Applications. John Wiley, Chichester.Google Scholar
Pitman, E. J. G. (1937). The “closest” estimates of statistical parameters. Proc. Camb. Phil. Soc. 33, 212–222.Google Scholar
Rosenblatt, M. (1965). Products of independent and identically distributed stochastic matrices J. Math. Anal. Appl. 11, 1–10.Google Scholar
Stoyanov, J. and Pirinsky, C (2000). Random motions, classes of ergodic Markov chains and beta distributions. Statist. Prob. Lett. 50, 293–304.CrossRefGoogle Scholar
Touri, B. (2012). Product of Random Stochastic Matrices and Distributed Averaging. Springer, Berlin.Google Scholar
Van Assche, W. (1986). Products of 2 × 2 stochastic matrices with random entries. J. Appl. Prob. 23, 1019–1024.Google Scholar
Volodin, N. A., Kotz, S. and N. L., Johnson. (1993). Use of moments in distribution theory: a multivariate case. J. Multivariate Anal. 46, 112–119.Google Scholar