aequationes mathematicae

, Volume 49, Issue 1, pp 57–85 | Cite as

Local limit approximations for Lagrangian distributions

  • Ljuben Mutafchiev
Research Papers


The discrete probability distribution function
$$\Pr (Y = m) = \frac{1}{{m!}}\frac{{d^{m - 1} }}{{dz^{m - 1} }}(f (z)g^m (z))|_{z = 0} ,m = 1,2,...,$$
and Pr(Y = 0) =f(0) define the class of the Lagrangian distributions iff andg are arbitrary probability generating functions of random variables on 0, 1, ⋯ andg(0) ≠ 0. In this paper we consider the case wheref andg are generating functions of power series distributed random variables. More precisely, we suppose thatf(z) = φ(λz)/φ(λ) andg(z) = ψ(μz)/ψ(μ), whereø andψ are power series with non-negative coefficients such thatφ diverges at its radius of convergence,ψ(0) ≠ 0 andλ andμ are certain parameters. Under some assumptions onφ andψ of admissibility type, various local limit theorems are proved for this class of Lagrangian distributions asm → ∞ and the parametersλ andμ change in an appropriate way. Finally, several approximations of well-known Lagrangian distributions (Lagrangian Poisson, Lagrangian binomial, generalized negative binomial, etc.) are demonstrated.

AMS (1991) Classification

Primary 69F05 Secondary 05A16, 60C05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Consul, P. C.,Lagrange and related probability distributions. InEncyclopedia of statistical sciences, Vol. 4 (S. Kotz, N. Johnson, eds.), John Wiley and Sons, New York, 1983, pp. 448–454.Google Scholar
  2. [2]
    Consul, P. C. andShenton, L. R.,Use of Lagrange expansion for generating generalized probability distributions. SIAM J. Appl. Math.23 (1972), 239–248.CrossRefGoogle Scholar
  3. [3]
    Consul, P. C. andShenton, L. R.,On the probabilistic structure and properties of discrete Lagrangian distributions. InStatistical distributions in scientific work, Vol. 1 (C. Taillie, G. P. Patil, and B. Baldessari, eds.), D. Reidel, Dordrecht-Holland, 1974, pp. 41–57.Google Scholar
  4. [4]
    De Bruijn, N. G.,Asymptotic methods in analysis. North Holland, Amsterdam, 1958.Google Scholar
  5. [5]
    Good, I. J.,The generalization of Lagrange's expansion and the enumeration of trees. Proc. Cambridge Philos. Soc.61 (1965), 499–517.Google Scholar
  6. [6]
    Good, I. J.,The Lagrange distributions and branching processes. SIAM J. Appl. Math.28 (1975), 270–275.CrossRefGoogle Scholar
  7. [7]
    Hayman, W. K.,A generalization of Stirling's formula. J. Reine Angew. Math.196 (1956), 67–95.Google Scholar
  8. [8]
    Hurwitz, A. andCourant, R.,Allgemeine Funktionentheorie und elliptische Funktionen. Geometrische Funktionentheorie. Springer-Verlag, Berlin, 1964.Google Scholar
  9. [9]
    Kolchin, V. F.,Random mappings. Optimization Software, New York, 1986.Google Scholar
  10. [10]
    Loéve, M.,Probability theory (2nd ed.). Van Nostrand, Princeton, 1960.Google Scholar
  11. [11]
    Lukacs, E.,Characteristic functions. Griffin, London, 1970.Google Scholar
  12. [12]
    Meir, A. andMoon, J. W.,On the altitude of nodes in random trees. Canad. J. Math.30 (1978), 997–1015.Google Scholar
  13. [13]
    Meir, A. andMoon, J. W.,The asymptotic behaviour of coefficients of powers in certain generating functions. European J. Combin.11 (1990), 581–587.Google Scholar
  14. [14]
    Meir, A., Moon, J. W. andMycielski, J.,Hereditarily finite sets and identity trees. J. Combin. Theory Ser. B35 (1983), 142–155.CrossRefGoogle Scholar
  15. [15]
    Mutafchiev, L. R.,Local limit theorems for sums of power series distributed random variables and for the number of components in labelled relational structures. Random Structures Algorithms3 (1992), 403–426.Google Scholar
  16. [16]
    Odlyzko, A. M. andRichmond, L. B.,Asymptotic expansions for the coefficients of analytic generating functions. Aequationes Math.28 (1985), 50–63.Google Scholar
  17. [17]
    Riordan, J.,Combinatorial identities. John Wiley, New York, 1968.Google Scholar
  18. [18]
    Steyaert, J.-M. andFlajolet, P.,Patterns and pattern-matching in trees:an analysis. Inform. and Control58 (1983), 19–58.CrossRefGoogle Scholar
  19. [19]
    Stromberg, K. R.,Introduction to classical real analysis. Wadsworth, New York, 1981.Google Scholar

Copyright information

© Birkhäuser Verlag 1995

Authors and Affiliations

  • Ljuben Mutafchiev
    • 1
  1. 1.Institute of MathematicsBulgarian Academy of SciencesSofiaBulgaria

Personalised recommendations