A combinatorial identity for a problem in asymptotic statistics

H. Albrecher, J.L. Teugels, K. Scheicher

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Abstract

Let (Xi)i=1 be a sequence of positive independent identically distributed random variables with regularly varying distribution tail of index 0 <a <1 and define Tn = X1²+X2²+···+Xn²/(X1+X2+···+ Xn)².In this note we simplify an expression for lim n¿8 E(T kn ), which was obtained by Albrecher and Teugels: Asymptotic analysis of a measure of variation. Theory Prob. Math. Stat., 74 (2006), 1-9, in terms of coefficients of a continued fraction expansion. The new formula establishes an unexpected link to an enumeration problem for rooted maps on orientable surfaces that was studied in Arquès and Béraud: Rooted maps of orientable surfaces, Riccati's equation and continued fractions.
Original languageEnglish
Pages (from-to)64-68
Number of pages5
JournalApplicable Analysis and Discrete Mathematics
Volume3
Issue number1
DOIs
Publication statusPublished - 2009

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