## The Annals of Applied Probability

### A central limit theorem for stochastic recursive sequences of topical operators

Glenn Merlet

#### Abstract

Let (An)n∈ℕ be a stationary sequence of topical (i.e., isotone and additively homogeneous) operators. Let x(n, x0) be defined by x(0, x0)=x0 and x(n+1, x0)=Anx(n, x0). It can model a wide range of systems including train or queuing networks, job-shop, timed digital circuits or parallel processing systems.

When (An)n∈ℕ has the memory loss property, (x(n, x0))n∈ℕ satisfies a strong law of large numbers. We show that it also satisfies the CLT if (An)n∈ℕ fulfills the same mixing and integrability assumptions that ensure the CLT for a sum of real variables in the results by P. Billingsley and I. Ibragimov.

#### Article information

Source
Ann. Appl. Probab. Volume 17, Number 4 (2007), 1347-1361.

Dates
First available in Project Euclid: 10 August 2007

https://projecteuclid.org/euclid.aoap/1186755242

Digital Object Identifier
doi:10.1214/105051607000000168

Mathematical Reviews number (MathSciNet)
MR2344309

Zentralblatt MATH identifier
1220.60016

#### Citation

Merlet, Glenn. A central limit theorem for stochastic recursive sequences of topical operators. Ann. Appl. Probab. 17 (2007), no. 4, 1347--1361. doi:10.1214/105051607000000168. https://projecteuclid.org/euclid.aoap/1186755242

#### References

• Baccelli, F. (1992). Ergodic theory of stochastic Petri networks. Ann. Probab. 20 375--396.
• Baccelli, F., Cohen, G., Olsder, G. J. and Quadrat, J. P. (1992). Synchronisation and Linearity. Wiley, New York.
• Baccelli, F. and Hong, D. (2000). Analytic expansions of max-plus Lyapunov exponents. Ann. Appl. Probab. 10 779--827.
• Baccelli, F. and Hong, D. (2000). Analyticity of iterates of random non-expansive maps. Adv. in Appl. Probab. 32 193--220.
• Baccelli, F. and Mairesse, J. (1998). Ergodic theorems for stochastic operators and discrete event networks. In Idempotency (Bristol, 1994) 171--208. Cambridge Univ. Press.
• Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
• Blondel, V. D., Gaubert, S. and Tsitsiklis, J. N. (2000). Approximating the spectral radius of sets of matrices in the max-algebra is NP-hard. IEEE Trans. Automat. Control 45 1762--1765.
• Bousch, T. and Mairesse, J. (2006). Finite-range topical functions and uniformly topical functions. Dyn. Syst. 21 73--114.
• Braker, H. (1993). Algorithms and applications in timed discrete event systems. Ph.D. thesis, Delft Univ. Technology.
• Chazottes, J.-R. (2003). Hitting and returning to non-rare events in mixing dynamical systems. Nonlinearity 16 1017--1034.
• Cohen, G., Dubois, D., Quadrat, J. P. and Viot, M. (1985). A linear system theoretic view of discrete event processes and its use for performance evaluation in manufacturing. IEEE Trans. Automat. Control AC--30 210--220.
• Cohen, J. E. (1988). Subadditivity, generalized products of random matrices and operations research. SIAM Rev. 30 69--86.
• Crandall, M. G. and Tartar, L. (1980). Some relations between nonexpansive and order preserving mappings. Proc. Amer. Math. Soc. 78 385--390.
• Gaubert, S. and Hong, D. (2000). Series expansions of Lyapunov exponents and forgetful monoids. Technical report, INRIA.
• Gaubert, S. and Mairesse, J. (1999). Modeling and analysis of timed Petri nets using heaps of pieces. IEEE Trans. Automat. Control 44 683--697.
• Gaubert, S. and Mairesse, J. (1998). Task resource models and $(\max,+)$ automata. In Idempotency (Bristol, 1994) 133--144. Cambridge Univ. Press.
• Gaujal, B. and Jean-Marie, A. (1998). Computational issues in recursive stochastic systems. In Idempotency (Bristol, 1994) 209--230. Cambridge Univ. Press.
• Gunawardena, J. and Keane, M. (1995). The existence of cycle times for some nonexpansive maps. Technical Report HPL-BRIMS-95-003, Hewlett-Packard Labs.
• Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and Its Application. Academic Press Inc, New York.
• Heidergott, B. (2000). A characterisation of $(\max,+)$-linear queueing systems. Queueing Syst. Theory Appl. 35 237--262.
• Heidergott, B. and de Vries, R. (2001). Towards a $(\operatornameMax,+)$ control theory for public transportation networks. Discrete Event Dyn. Syst. 11 371--398.
• Heidergott, B., Oldser, G. J. and van der Woude, J. (2006). Max Plus at Work. Princeton Univ. Press.
• Hennion, H. (1997). Limit theorems for products of positive random matrices. Ann. Probab. 25 1545--1587.
• Ibragimov, I. A. and Linnik, Yu. V. (1971). Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff Publishing, Groningen.
• Ishitani, H. (1976/77). A central limit theorem for the subadditive process and its application to products of random matrices. Publ. Res. Inst. Math. Sci. 12 565--575.
• Mairesse, J. (1997). Products of irreducible random matrices in the $(\max,+)$ algebra. Adv. in Appl. Probab. 29 444--477.
• Merlet, G. (2004). Memory loss property for products of random matrices in the $(\max,+)$ algebra. Technical report, IRMAR. Available at http://hal.ccsd.cnrs.fr/ccsd-00001607.
• Merlet, G. (2005). Limit theorems for iterated random topical operators. Technical report, IRMAR. Available at http://hal.ccsd.cnrs.fr/ccsd-00004594.
• Merlet, G. (2005). Produits de matrices aléatoires: Exposants de Lyapunov pour des matrices aléatoires suivant une mesure de Gibbs, théorèmes limites pour des produits au sens max-plus. Ph.D. thesis, Univ. Rennes. Available at http://tel.ccsd.cnrs.fr/tel-00010813.
• Resing, J. A. C., de Vries, R. E., Hooghiemstra, G., Keane, M. S. and Olsder, G. J. (1990). Asymptotic behavior of random discrete event systems. Stochastic Process. Appl. 36 195--216.
• Vincent, J.-M. (1997). Some ergodic results on stochastic iterative discrete events systems. Discrete Event Dynamic Systems 7 209--232.