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The asymptotic distributions of generalized U-statistics with applications to random graphs
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  • Published: September 1991

The asymptotic distributions of generalized U-statistics with applications to random graphs

  • Svante Janson1 &
  • Krzysztof Nowicki2 

Probability Theory and Related Fields volume 90, pages 341–375 (1991)Cite this article

Summary

We consider the random variable

$$S_{n,v} (f) = \sum\limits_{i_1< ...< i_v \leqq n} {f(X_{i_1 } ,...,X_{i_v } ,Y_{i_1 i_2 } ,...,Y_{i_{v - 1} i_v } ),}$$

where {X i } =1/n i and {Y ij }1≦i<i≦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 formS n, v (f). We use a projection method and expandf into a (finite) sum of terms of increasing complexity. The terms in such an expansion are indexed by graphs and the asymptotic behaviour ofS n, v 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, v (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.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Uppsala University, Thunbergsv. 3, S-752 38, Uppsala, Sweden

    Svante Janson

  2. Department of Statistics, University of Lund, Box 7008, S-220 07, Lund, Sweden

    Krzysztof Nowicki

Authors
  1. Svante Janson
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  2. Krzysztof Nowicki
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Janson, S., Nowicki, K. The asymptotic distributions of generalized U-statistics with applications to random graphs. Probab. Th. Rel. Fields 90, 341–375 (1991). https://doi.org/10.1007/BF01193750

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  • Received: 07 August 1989

  • Issue Date: September 1991

  • DOI: https://doi.org/10.1007/BF01193750

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Keywords

  • Stochastic Process
  • Asymptotic Behaviour
  • Probability Theory
  • Mathematical Biology
  • General Treatment
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