The Annals of Applied Probability

A survey of max-type recursive distributional equations

David J. Aldous and Antar Bandyopadhyay

Full-text: Open access


In certain problems in a variety of applied probability settings (from probabilistic analysis of algorithms to statistical physics), the central requirement is to solve a recursive distributional equation of the form $X\mathop{=}\limits^{d}\,g((\xi_{i},X_{i}),i\geq 1)$. Here (ξi) and g(⋅) are given and the Xi are independent copies of the unknown distribution X. We survey this area, emphasizing examples where the function g(⋅) is essentially a “maximum” or “minimum” function. We draw attention to the theoretical question of endogeny: in the associated recursive tree process Xi, are the Xi measurable functions of the innovations process (ξi)?

Article information

Ann. Appl. Probab. Volume 15, Number 2 (2005), 1047-1110.

First available in Project Euclid: 3 May 2005

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60E05: Distributions: general theory 62E10: Characterization and structure theory 68Q25: Analysis of algorithms and problem complexity [See also 68W40] 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

Branching process branching random walk cavity method coupling from the past fixed point equation frozen percolation mean-field model of distance metric contraction probabilistic analysis of algorithms probability distribution probability on trees random matching


Aldous, David J.; Bandyopadhyay, Antar. A survey of max-type recursive distributional equations. Ann. Appl. Probab. 15 (2005), no. 2, 1047--1110. doi:10.1214/105051605000000142.

Export citation


  • Aldous, D. J. (1992). Asymptotics in the random assignment problem. Probab. Theory Related Fields 93 507--534.
  • Aldous, D. J. (1992). Greedy search on the binary tree with random edge-weights. Combin. Probab. Comput. 1 281--293.
  • Aldous, D. J. (1998). A Metropolis-type optimization algorithm on the infinite tree. Algorithmica 22 388--412.
  • Aldous, D. J. (1998). On the critical value for percolation of minimum-weight trees in the mean-field distance model. Combin. Probab. Comput. 7 1--10.
  • Aldous, D. J. (2000). The percolation process on a tree where infinite clusters are frozen. Math. Proc. Cambridge Philos. Soc. 128 465--477.
  • Aldous, D. J. (2001). The $\zeta(2)$ limit in the random assignment problem. Random Structures Algorithms 18 381--418.
  • Aldous, D. J. (2004). Cost-volume relationship for flows through a disordered network. Unpublished manuscript.
  • Aldous, D. J. (2005). Percolation-like scaling exponents for minimal paths and trees in the stochastic mean-field model. Proc. Roy. Soc. London Ser. A Math. Phys. Eng. Sci. To appear.
  • Aldous, D. J. and Krebs, W. B. (1990). The birth-and-assassination process. Statist. Probab. Lett. 10 427--430.
  • Aldous, D. J. and Percus, A. G. (2003). Scaling and universality in continuous length combinatorial optimization. Proc. Natl. Acad. Sci. U.S.A. 100 11211--11215.
  • Aldous, D. J. and Steele, J. M. (2003). The objective method: Probabilistic combinatorial optimization and local weak convergence. In Probability on Discrete Structures, Encyclopaedia of Mathematical Sciences 110 (H. Kesten, ed.) 1--72. Springer, New York.
  • Asmussen, S. (1987). Applied Probability and Queues. Wiley, New York.
  • Athreya, K. B. (1985). Discounted branching random walks. Adv. in Appl. Probab. 17 53--66.
  • Bachmann, M. (2000). Limit theorems for the minimal position in a branching random walk with independent logconcave displacements. Adv. in Appl. Probab. 32 159--176.
  • Bandyopadhyay, A. (2002). Bivariate uniqueness in the logistic fixed point equation. Technical Report 629, Dept. Statistics, Univ. California, Berkeley.
  • Bandyopadhyay, A. (2004). Bivariate uniqueness and endogeny for recursive distributional equations: Two examples. Preprint.
  • Barlow, M. T., Pemantle, R. and Perkins, E. (1997). Diffusion-limited aggregation on a tree. Probab. Theory Related Fields 107 1--60.
  • Bertoin, J. (2003). The asymptotic behaviour of fragmentation processes. J. European Math. Soc. 5 395--416.
  • Biggins, J. D. (1977). Chernoff's theorem in the branching random walk. J. Appl. Probab. 14 630--636.
  • Biggins, J. D. (1977). Martingale convergence in the branching random walk. J. Appl. Probab. 14 25--37.
  • Biggins, J. D. (1998). Lindley-type equations in the branching random walk. Stochastic Process. Appl. 75 105--133.
  • Caliebe, A. and Rösler, U. (2003). Fixed points with finite variance of a smoothing transformation. Stochastic Process. Appl. 107 105--129.
  • Chauvin, B., Krein, T., Marckert, J.-F. and Rouault, A. (2004). Martingales and profiles of binary search trees. Available at
  • Dekking, F. M. and Host, B. (1991). Limit distributions for minima displacements of branching random walks. Probab. Theory Related Fields 90 403--426.
  • Devroye, L. (2001). On the probabilistic worst-case time of find. Algorithmica 31 291--303.
  • Devroye, L. and Neininger, R. (2002). Density approximation and exact simulation of random variables that are solutions of fixed-point equations. Adv. in Appl. Probab. 34 441--468.
  • Diaconis, P. and Freedman, D. (1999). Iterated random functions. SIAM Review 41 45--76.
  • Durrett, R. and Liggett, T. M. (1983). Fixed points of the smoothing transformation. Z. Wahrsch. Verw. Gebiete 64 275--301.
  • Durrett, R. and Limic, V. (2002). A surprising Poisson process arising from a species competition model. Stochastic Process. Appl. 102 301--309.
  • Elias, P. (1972). The efficient construction of an unbiased random source. Ann. Math. Statist. 43 865--870.
  • Fill, J. A. and Janson, S. (2000). A characterization of the set of fixed points of the quicksort transformation. Electron. Comm. Probab. 5 77--84.
  • Frieze, A. M. (1985). On the value of a random minimum spanning tree problem. Discrete Appl. Math. 10 47--56.
  • Gamarnik, D., Nowicki, T. and Swirscsz, G. (2003). Maximum weight independent sets and matchings in sparse random graphs: Exact results using the local weak convergence method. arXiv:math.PR/0309441.
  • Georgii, H.-O. (1988). Gibbs Measures and Phase Transitions. de Gruyter, Berlin.
  • Grimmett, G. R. (1999). Percolation, 2nd ed. Springer, Berlin.
  • Hammersley, J. M. (1974). Postulates for subadditive processes. Ann. Probab. 2 652--680.
  • Harris, S. C. (1999). Travelling-waves for the FKPP equation via probabilistic arguments. Proc. Roy. Soc. Edinburgh Sect. A 129 503--517.
  • Iksanov, A. M. (2004). Elementary fixed points of the BRW smoothing transforms with infinite number of summands. Stochastic Process. Appl. 114 27--50.
  • Iksanov, A. M. and Jurek, Z. J. (2002). On fixed points of Poisson shot noise transforms. J. Appl. Probab. 34 798--825.
  • Iksanov, A. M. and Kim, C.-S. (2004). On a Pitman--Yor problem. Statist. Probab. Lett. 68 61--72.
  • Jagers, P. and Rösler, U. (2004). Stochastic fixed points for the maximum. In Mathematics and Computer Science III (M. Drmota, P. Flajolet, D. Gardy and B. Gittenberger, eds.) 325--338. Birkhäuser, Basel.
  • Jordan, J. (2003). Renormalization of random hierarchical systems. Ph.D. thesis, Univ. Sheffield, U.K. Available at
  • Kagan, A. M., Linnik, Yu. V. and Rao, C. R. (1973). Characterization Problems in Mathematical Statistics. Wiley, New York.
  • Karpelevich, F. I., Kelbert, M. Ya. and Suhov, Yu. M. (1994). Higher-order Lindley equations. Stochastic Process. Appl. 53 65--96.
  • Khamsi, M. A. and Kirk, W. A. (2001). An Introduction to Metric Spaces and Fixed Point Theory. Wiley, New York.
  • Kolmogorov, A., Petrovsky, I. and Piscounov, N. (1937). Etude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique. Moscow Univ. Math. Bull. 1 1--25.
  • Liggett, T. M. (1985). Interacting Particle Systems. Springer, Berlin.
  • Linusson, S. and Wästlund, J. (2004). A proof of Parisi's conjecture on the random assignment problem. Probab. Theory Related Fields 128 419--440.
  • Liu, Q. (1998). Fixed points of a generalized smoothing transformation and applications to the branching random walk. Adv. in Appl. Probab. 30 85--112.
  • Liu, Q. (2001). Asymptotic properties and absolute continuity of laws stable by random weighted mean. Stochastic Process. Appl. 95 83--107.
  • Mézard, M. and Parisi, G. (1985). Replicas and optimization. J. Physique Lett. 46 L771--L778.
  • Mézard, M. and Parisi, G. (1986). A replica analysis of the travelling salesman problem. J. Physique 47 1285--1296.
  • Mézard, M. and Parisi, G. (1987). On the solution of the random link matching problem. J. Physique 48 1451--1459.
  • Mézard, M. and Parisi, G. (2003). The cavity method at zero temperature. J. Statist. Phys. 111 1--34.
  • Nair, C., Prabhakar, B. and Sharma, M. (2003). A proof of Parisi's conjecture for the finite random assignment problem. Unpublished manuscript.
  • Propp, J. and Wilson, D. (1998). Coupling from the past: A user's guide. In Microsurveys in Discrete Probability. DIMACS Ser. Discrete Math. Theoret. Comp. Sci. (D. Aldous and J. Propp, eds.) 41 181--192.
  • Rachev, S. T. (1991). Probability Metrics and the Theory of Stochastic Models. Wiley, New York.
  • Rachev, S. T. and Rüschendorf, L. (1998). Mass Transportation Problems 2: Applications. Springer, New York.
  • Rösler, U. (1992). A fixed point theorem for distributions. Stochastic Process. Appl. 42 195--214.
  • Rösler, U. and Rüschendorf, L. (2001). The contraction method for recursive algorithms. Algorithmica 29 3--33.
  • Steele, J. M. (1997). Probability Theory and Combinatorial Optimization. SIAM, Philadelphia, PA.
  • Talagrand, M. (2003). An assignment problem at high temperature. Ann. Probab. 31 818--848.
  • Uchaikin, V. V. and Zolotarev, V. M. (1999). Chance and Stability. VSP, Utrecht.