aequationes mathematicae

, Volume 49, Issue 1, pp 57–85

Local limit approximations for Lagrangian distributions

• Ljuben Mutafchiev
Research Papers

Summary

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

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