# A combinatorial identity for a problem in asymptotic statistics

### Details

Serval ID

serval:BIB_64EBE236A40B

Type

**Article**: article from journal or magazin.

Collection

Publications

Fund

Title

A combinatorial identity for a problem in asymptotic statistics

Journal

Applicable Analysis and Discrete Mathematics

ISSN

1452-8630

Publication state

Published

Issued date

2009

Peer-reviewed

Oui

Volume

3

Number

1

Pages

64-68

Language

english

Abstract

Let (X-i)(i >= 1) be a sequence of positive independent identically distributed random variables with regularly varying distribution tail of index 0 < alpha < 1 and define

T-n:= X-1(2) + X-2(2) + ... + X-n(2) / (X-1 + X-2 + ... + X-n)(2).

In this note we simplify an expression for (n ->infinity) lim E(T-n(k)), which was obtained by ALBRECHER and TEUGELS: A symptotic 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 ARQUES and BERAUD: Rooted maps of orientable surfaces, Riccati's equation and continued fractions. Discrete Mathematics, 215 (2000), 1-12.

T-n:= X-1(2) + X-2(2) + ... + X-n(2) / (X-1 + X-2 + ... + X-n)(2).

In this note we simplify an expression for (n ->infinity) lim E(T-n(k)), which was obtained by ALBRECHER and TEUGELS: A symptotic 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 ARQUES and BERAUD: Rooted maps of orientable surfaces, Riccati's equation and continued fractions. Discrete Mathematics, 215 (2000), 1-12.

Keywords

Asymptotic behavior, Generating functions, Continued fraction, Regularly varying functions, Enumeration problems

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Create date

09/02/2009 18:42

Last modification date

03/03/2018 16:50