A COMBINATORIAL IDENTITY FOR A PROBLEM IN ASYMPTOTIC STATISTICS

Let (Xi)i≥1 be a sequence of positive independent identically distributed random variables with regularly varying distribution tail of index 0 < α < 1 and define ${T_n}\,: = \,\frac{{X_1^2\, + \,X_2^2\, + \,\cdot\cdot\cdot\, + \,X_n^2}}{{{\left( {{X_1}\, + \,{X_2}\, + \,\cdot\cdot\cdot\, + \,{X_n}} \right)}^2}}}$ In this note we simplify an expression for $\mathop {\lim }\limits_{n \to \infty } \,\mathbb{E}\left( {T_n^k} \right)$, 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. Discrete Mathematics, 215 (2000), 1-12.