The Annals of Applied Probability

Large deviations of combinatorial distributions. II. Local limit theorems

Hsien-Kuei Hwang

Full-text: Open access

Abstract

We derive a general local limit theorem for probabilities of large deviations for a sequence of random variables by means of the saddlepoint method on Laplace-type integrals. This result is applicable to parameters in a number of combinatorial structures and the distribution of additive arithmetical functions.

Article information

Source
Ann. Appl. Probab., Volume 8, Number 1 (1998), 163-181.

Dates
First available in Project Euclid: 29 July 2002

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1027961038

Digital Object Identifier
doi:10.1214/aoap/1027961038

Mathematical Reviews number (MathSciNet)
MR1620350

Zentralblatt MATH identifier
0954.60020

Subjects
Primary: 60F10: Large deviations
Secondary: 05A16: Asymptotic enumeration 11K65: Arithmetic functions [See also 11Nxx]

Keywords
Large deviations local limit theorems asymptotic expansion saddle-point method combinatorial schemes singularity analysis additive arithmetical functions

Citation

Hwang, Hsien-Kuei. Large deviations of combinatorial distributions. II. Local limit theorems. Ann. Appl. Probab. 8 (1998), no. 1, 163--181. doi:10.1214/aoap/1027961038. https://projecteuclid.org/euclid.aoap/1027961038


Export citation

References

  • [1] Arratia, R., Stark, D. and Tavar´e, S. (1995). Total variation asy mptotics for Poisson process approximations of logarithmic combinatorial assemblies. Ann. Probab. 23 1347- 1388.
  • [2] Bender, E. A. (1973). Central and local limit theorems applied to asy mptotic enumeration. J. Combin. Theory Ser. A 15 91-111.
  • [3] Canfield, E. R. (1977). Central and local limit theorems for the coefficients of poly nomials of binomial ty pe. J. Combin. Theory Ser. A 23 275-290.
  • [4] Chaganty, N. R. and Sethuraman, J. (1985). Large deviation local limit theorems for arbitrary sequences of random variables. Ann. Probab. 13 97-114.
  • [5] Comtet, L. (1974). Advanced Combinatorics, the Art of Finite and Infinite Expansions, rev. ed. Reidel, Dordrecht.
  • [6] Cram´er, H. (1970). Random Variables and Probability Distributions, 3rd ed. Cambridge Univ. Press.
  • [7] Drmota, M. (1994). A bivariate asy mptotic expansion of coefficients of powers of generating functions. European J. Combin. 15 139-152.
  • [8] Erd os, P. (1948). On the integers having exactly k prime factors. Ann. Math. 49 53-66.
  • [9] Flajolet, P. and Odly zko, A. M. (1990). Random mapping statistics. Lecture Notes in Comput. Sci. 434 329-354. Springer, Berlin.
  • [10] Flajolet, P. and Odly zko, A. M. (1990). Singularity analysis of generating functions. SIAM J. Discrete Math. 3 216-240.
  • [11] Flajolet, P. and Soria, M. (1990). Gaussian limiting distributions for the number of components in combinatorial structures. J. Combin. Theory Ser. A 53 165-182.
  • [12] Flajolet, P. and Soria, M. (1993). General combinatorial schemes, Gaussian limit distributions and exponential tails. Discrete Math. 114 159-180.
  • [13] Gao, Z. and Richmond, L. B. (1992). Central and local limit theorems applied to asy mptotic enumeration IV, multivariate generating functions. J. Comput. Appl. Math. 41 177-186.
  • [14] Greene, D. H. and Knuth, D. E. (1990). Mathematics for the Analy sis of Algorithms, 3rd ed. Birkh¨auser, Boston.
  • [15] Hansen, J. C. (1994). Order statistics for decomposable combinatorial structures. Random Structures Algorithms 5 517-533.
  • [16] Hwang, H.-K. (1994). Th´eor emes limites pour les structures combinatoires et les fonctions arithm´etiques. Ph.D dissertation, ´Ecole Poly technique, Palaiseau.
  • [17] Hwang, H.-K. (1995). Asy mptotic expansions for Stirling's number of the first kind. J. Combin. Theory Ser. A 71 343-351.
  • [18] Hwang, H.-K. (1996). Large deviations of combinatorial distributions I. Central limit theorems. Ann. Appl. Probab. 6 297-319.
  • [19] Hwang, H.-K. (1996). Convergence rates in the central limit theorems for combinatorial structures. European J. Combin. To appear.
  • [20] Jensen, J. L. (1995). Saddlepoint Approximations. Oxford Science Publishers.
  • [21] Kolchin, V. F. (1986). Random Mappings. Optimization Software, New York.
  • [22] Kubilius, J. (1964). Probabilistic Methods in the Theory of Numbers. Amer. Math. Soc., Providence, RI.
  • [23] Ledoux, M. (1992). Sur les d´eviations mod´er´ees des sommes de variables al´eatoires vectorielles ind´ependantes de m eme loi. Ann. Inst. H. Poincar´e Probab. Statist. 28 267-280.
  • [24] Meir, A. and Moon, J. W. (1990). The asy mptotic behaviour of coefficients of powers of certain generating functions. European J. Combin. 11 581-587.
  • [25] Norton, K. K. (1978). Estimates for partial sums of the exponential series. J. Math. Anal. Appl. 63 265-296.
  • [26] Otter, R. (1948). The number of trees. Ann. Math. 49 583-599.
  • [27] Pavlov, A. I. (1988). Local limit theorems for the number of components of random permutations and mappings. Theory Probab. Appl. 33 183-187.
  • [28] Richter, W. (1957). Local limit theorems for large deviations. Theory Probab. Appl. 2 206- 220.
  • [29] Selberg, A. (1954). Note on a paper by L. G. Sathe. J. Indian Math. Soc. 18 83-87.
  • [30] Tenenbaum, G. (1990). Introduction a la Th´eorie Analy tique et Probabiliste des Nombres. Institut Elie Cartan, Universit´e de Nancy I.