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Brownian bridge asymptotics for random mappings. (English) Zbl 0811.60057
This paper deals with the uniform model of random mappings of an \(n\)- element set into itself. The authors introduce a new technique, which starts by specifying a coding of mappings as walks with \(\pm 1\) steps. Then, the uniform random mapping model is thereby coded as a nonuniform random walk. The main result of the paper shows that as \(n\to\infty\) the random walk rescales to reflecting Brownian bridge. This enables to obtain a large number of limiting distributions concerning numerical characteristics of the random digraphs representing random mappings.

MSC:
60G50 Sums of independent random variables; random walks
60J65 Brownian motion
60C05 Combinatorial probability
60F05 Central limit and other weak theorems
05C80 Random graphs (graph-theoretic aspects)
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[1] Aldous, Lecture Notes in Mathematics 1117, in: École d’Été St. Flour 1983 pp 1– (1985)
[2] Aldous, Stochastic Analysis pp 23– (1991)
[3] Aldous, The continuum random tree III, Ann. Probab. 21 pp 248– (1983)
[4] D. J. Aldous J. Pitman Distributional aspects of a recursive decomposition of Brownian bridge and random mapping asymptotics, in preparation 1994
[5] Bagaev, Limit distributions of metric characteristics of an indecomposable random mapping, Combin. Asymp. Anal. Krasnojarsk. Gos. Univ. 2 pp 55– (1977)
[6] Bertoin, Path transformations connecting Brownian bridge, excursion and meander, Bull. Sci. Math. (1994) · Zbl 0805.60076
[7] Bhattacharaya, Stochastic Processes with Applications (1990)
[8] Billingsley, Convergence of Probability Measures (1968)
[9] Chung, Excursions in Brownian motion, Ark. Mat. 14 pp 155– (1976) · Zbl 0356.60033
[10] Ethier, Markov Processes: Characterization and Convergence (1986) · doi:10.1002/9780470316658
[11] Flajolet, Lecture Notes in Computer Science 434, in: Advances in Cryptology-EUROCRYPT ’89 pp 329– (1990)
[12] D. Foata La Série Génératrice Exponentielle dans les Problèmes d’Enumération 1974
[13] Graf, The exact Hausdorff dimension in random recursive constructions, Mem. Am. Math. Soc. 71 (381) pp 1– (1988)
[14] Hansen, A functional central limit theorem for random mappings, Ann. Probab. 17 pp 317– (1989) · Zbl 0667.60009
[15] Ann. Probab. 19 pp 1393– (1991)
[16] Ibragimov, Independent and Stationary Sequences of Random Variables (1971)
[17] Imhof, On Brownian bridge and excursion, Stud. Sci. Math. Hung. 20 pp 1– (1985) · Zbl 0625.60096
[18] Itǒ, Diffusion Processes and Their Sample Paths (1965) · doi:10.1007/978-3-642-62025-6
[19] Johnson, An explicit formula for the c.d.f. of the L1 norm of the Brownian bridge, Ann. Probab. 11 pp 807– (1983) · Zbl 0516.60044
[20] Kallenberg, Canonical representations and convergence criteria for processes with interchangeable increments, Z. Wahrsch. Verw. Gebiete 27 pp 23– (1973) · Zbl 0253.60060
[21] Knight, Essentials of Brownian Motion and Diffusion RI (1981) · doi:10.1090/surv/018
[22] Kolchin, Random Mappings (1986)
[23] Lévy, Sur certains processus stochastiques homogènes, Comp. Math. 7 pp 283– (1939) · JFM 65.1346.02
[24] L. Mutafciev Probability distributions and asymptotics for some characteristics of radom mappings 1983 227 238
[25] L. Mutafciev On some stochastic problems of discrete mathematics Mathematics and Education: Proc. 13th Spring Conf. Bulgarian Mathematicians 1984 57 80
[26] Perman, Order statistics for jumps of normalized subordinators, Stoch. Proc. Appl. 46 pp 267– (1993) · Zbl 0777.60070
[27] Perman, Size-biased sampling of Poisson point processes and excursions, Probab. Theor. Rel. Fields 92 pp 21– (1992) · Zbl 0741.60037
[28] J. Pitman Distribution of local times of Brownian bridge, unpublished 1991 · Zbl 0945.60081
[29] Proskurin, On the distribution of the number of vertices in strata of a random mapping, Theory Probab. Appl. 18 pp 803– (1973) · Zbl 0324.60009
[30] Revuz, Continuous Martingales and Brownian Motion (1991) · doi:10.1007/978-3-662-21726-9
[31] Salisbury, Construction of right processes from excursions, Z. Wahrsch. Verw. Gebiete 73 pp 351– (1986) · Zbl 0587.60068
[32] Stepanov, Limit distributions of certain characteristics of random mappings, Theory Probab. Appl. 14 pp 612– (1969)
[33] Stepanov, Random mappings with a single attracting center, Theory Probab. Appl. 16 pp 155– (1971) · Zbl 0239.60017
[34] Takacs, Random walk processes and their application in order statistics, Ann. Appl. Probab. 2 pp 435– (1992)
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