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Asymptotic distributions and a multivariate Darboux method in enumeration problems. (English) Zbl 0801.60016
Summary: Let $$c(x,z) = \sum c_{nk} x^ nz^ k$$ $$(c_{nk} \geq 0)$$ be a bivariate generating function satisfying a functional equation $$c = G(c,x,z)$$. By using a central limit theorem of Bender it is shown that discrete random variables $$X_ n$$ with $$P[X_ n=k] = c_{nk}/(\sum c_{ni})$$ are asymptotically normal with mean $$\mu_ n \sim \mu n$$ and variance $$\sigma^ 2_ n \sim \sigma^ 2n$$. Furthermore a bivariate asymptotic expansion for the coefficients $$c_{nk}$$ can be obtained by two different methods. After some applications to tree enumeration problems a multivariate Darboux-method is formulated.

##### MSC:
 60F05 Central limit and other weak theorems 60C05 Combinatorial probability
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##### References:
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