The asymptotic distributions of generalized U-statistics with applications to random graphs
type: Item
instance of: scholarly article
title: The asymptotic distributions of generalized U-statistics with applications to random graphs
zbMATH Open document ID: 0734.60036
DOI: 10.1007/BF01193750
author: Svante Janson
author: Krzysztof Nowicki
publication date: 1991-01-01T00:00:00Z
review text: We consider the random variable \[ S_{n,\nu}(f)=\sum_{i_ 1<...<i_{\nu}\leq n}f(X_{i_ 1},...,X_{i_{\nu}},Y_{i_ 1i_ 2},...,Y_{i_{\nu -1}i_{\nu}}), \] where \(\{X_ i\}^ n_{i=1}\) and \(\{Y_{ij}\}_{1\leq i<j\leq n}\) are two independent sequences of i.i.d. random variables. This paper gives a general treatment of the asymptotic behaviour of sums of the form \(S_{n,\nu}(f)\). We use a projection method and expand f into a (finite) sum of terms of increasing complexity. The terms in such an expansion are indexed by graphs and the asymptotic behaviour of \(S_{n,\nu}\) depends only on the non-zero terms indexed by the smallest graphs. Moreover, the type of the limit distribution depends on the topology of the graphs. In particular, \(S_{n,\nu}(f)\) is asymptotically normally distributed if these graphs are connected, but not otherwise. The general theorems are applied to the problems of finding the asymptotic distribution of the number of copies or induced copies of a given graph in various random graph models.
Mathematics Subject Classification ID: 60F99
Mathematics Subject Classification ID: 05C80
zbMATH DE Number: 4215060
zbMATH Keywords: asymptotic behaviour
zbMATH Keywords: topology of the graphs
zbMATH Keywords: random graph models
MaRDI profile type: Publication
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published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 4215060