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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.

60F05 Central limit and other weak theorems
60C05 Combinatorial probability
Full Text: DOI
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