The Annals of Applied Probability

Large Deviations of Products of Random Topical Operators

Fergal Toomey

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A topical operator on $\mathbb{R}^d$ is one which is isotone and homogeneous. Let ${A(n) : n \geq 1}$ be a sequence of i.i.d. random topical operators such that the projective radius of $A(n) \dots A(1)$ is almost surely bounded for large $n$. If ${x(n) : n \geq 1}$, is a sequence of vectors given by $x(n) = A(n) \dots A(1)x_0$, for some fixed initial condition $x_0$, then the sequence ${x(n)/n : n \geq 1}$ satisfies a weak large deviation principle. As corollaries of this result we obtain large deviation principles for products of certain random aperiodic max-plus and min-plus matrix operators and for products of certain random aperiodic nonnegative matrix operators.

Article information

Ann. Appl. Probab., Volume 12, Number 1 (2002), 317-333.

First available in Project Euclid: 12 March 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F10: Large deviations
Secondary: 47H40: Random operators [See also 47B80, 60H25] 47H07: Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces

Topical operators discrete event systems max-plus algebra nonnegative matrices large deviations


Toomey, Fergal. Large Deviations of Products of Random Topical Operators. Ann. Appl. Probab. 12 (2002), no. 1, 317--333. doi:10.1214/aoap/1015961166.

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  • BACCELLI, F., G. COHEN, G. J. OLSDER and J.-P. QUADRAT (1992). Synchronization and Linearity. Wiley, New York.
  • BACCELLI, F. and T. KONSTANTOPOULOS (1991). Estimates of cycle times in stochastic Petri nets. Applied Stochastic Analysis. Lecture Notes in Control and Inform. Sci. 177 1-20. Springer, New York.
  • BACCELLI, F. and J. MAIRESSE (1996). Ergodic theorems for stochastic operators and discrete event networks. In Idempotency (J. Gunawardena, ed.). Cambridge Univ. Press.
  • CHANG, C.-S. (1996). On the exponentiality of stochastic linear systems under the max-plus algebra. IEEE Trans. Automat. Control 41 1182-1188.
  • CRANDALL, M. G. and L. TARTAR (1980). Some relations between nonexpansive and order preserving maps. Proc. Amer. Math. Soc. 78 385-390.
  • DEMBO, A. and O. ZEITOUNI (1998). Large Deviations Techniques and Applications. Springer, New York.
  • GAUBERT, S. and J. GUNAWARDENA (1998). A non-linear hierarchy for discrete event dynamical systems. In Proceedings of the Fourth Workshop on Discrete Event Systems, Cagliari, Italy. IEE, London.
  • GLASSERMAN, P. and D. D. YAO (1995). Stochastic vector difference equations with stationary coefficients. J. Appl. Probab. 32 851-866.
  • GUNAWARDENA, J. (1994). Min-max functions. Discrete Event Dynamic Systems 4 377-406.
  • GUNAWARDENA, J. (1996). An introduction to idempotency. In Idempotency (J. Gunawardena, ed.). Cambridge Univ. Press.
  • GUNAWARDENA, J. and M. KEANE (1995). On the existence of cycle times for some nonexpansive maps. Technical Report HPL-BRIMS-95-003, HP Laboratories.
  • LANFORD, O. E. (1973). Entropy and equilibrium states in classical statistical mechanics. Lecture Notes in Phys. 20 1-113. Springer, New York.
  • LEWIS, J. T. and C.-E. PFISTER (1995). Thermodynamic probability theory: some aspects of large deviations. Russian Math. Surveys 50 279-317.
  • LEWIS, J. T., C.-E. PFISTER and W. G. SULLIVAN (1994). Entropy, concentration of probability, and conditional limit theorems. Markov Processes and Related Fields 1 319-386.
  • MAIRESSE, J. (1997). Products of irreducible random matrices in the max-plus algebra. Adv. in Appl. Probab. 29 444-477.
  • OLSDER, G. J. (1991). Eigenvalues of dynamic max-min systems. Discrete Event Dynamic Systems 1 177-207.