Asymptotic estimates using probability
type: Item
instance of: scholarly article
title: Asymptotic estimates using probability
zbMATH Open document ID: 0916.05003
author: William Beckner
author: Amitai Regev
published in: Advances in Mathematics
publication date: 1999-03-07T00:00:00Z
review text: The authors use probabilistic arguments in the spirit of the De Moivre-Laplace central limit theorem to obtain asymptotic estimates for combinatorial sums. The main question under discussion is the following. For a given sequence \(\chi_n=\sum_{\lambda\vdash n}f(\lambda)\chi_{\lambda}\) of \(S_n\)-characters defined in terms of the associated Young diagrams and with known asymptotics of the degrees when \(n\) tends to infinity, increase the length of the columns of the Young diagrams by a fixed factor \(q\), i.e. replace the partitions \(\lambda=(\lambda_1,\ldots,\lambda_k)\) with the new partitions \(q\ast\lambda= (\lambda_1^q,\ldots,\lambda_k^q)= (\lambda_1,\ldots,\lambda_1,\ldots, \lambda_k,\ldots,\lambda_k)\). Then determine the changes that occur in the asymptotics of the degrees of the new sequence \(\chi_{qn}= \sum_{\lambda\vdash n}f(\lambda)\chi_{q\ast\lambda}\) of \(S_{qn}\)-characters. The authors determine such estimates for the so-called Young derived sequences of characters. As a result they obtain some rather interesting integration formulas. In particular, some of the integrals involved are extensions of Selberg and Dyson-Macdonald-Mehta type integrals.
Mathematics Subject Classification ID: 05A16
Mathematics Subject Classification ID: 60C05
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 05A15
Mathematics Subject Classification ID: 05A19
Mathematics Subject Classification ID: 05E10
zbMATH DE Number: 1228412
zbMATH Keywords: probabilistic arguments
zbMATH Keywords: central limit theorem
zbMATH Keywords: combinatorial sums
zbMATH Keywords: asymptotic estimates
zbMATH Keywords: Young derived character sequences
zbMATH Keywords: characters of symmetric groups
zbMATH Keywords: Young diagrams
reviewed by: Vesselin Drensky
MaRDI profile type: Publication
OpenAlex ID: W2064769647
cites work: Asymptotics of Combinatorial Sums and the Central Limit Theorem
cites work: Asymptotic values for degrees associated with strips of Young diagrams
cites work: Young-derived sequences of \(S_ n\)-characters
DOI: 10.1006/AIMA.1994.1503
Recommended article: Asymptotic Estimates of Elementary Probability Distributions
Recommended article: Asymptotic approximations for probability integrals
Recommended article: Q3349703
Recommended article: Q5434185
Recommended article: Asymptotic expansions for probability distributions
Recommended article: Q3996549
Recommended article: Asymptotic Statistics
Recommended article: Asymptotische Gesetƶe der Wahrscheinlichkeitsrechnung
description: scientific article; zbMATH DE number 1228412