Probability Theory and Related Fields

, Volume 90, Issue 3, pp 341–375 | Cite as

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

  • Svante Janson
  • Krzysztof Nowicki
Article
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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 } i =1/n 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.

Keywords

Stochastic Process Asymptotic Behaviour Probability Theory Mathematical Biology General Treatment 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • Svante Janson
    • 1
  • Krzysztof Nowicki
    • 2
  1. 1.Department of MathematicsUppsala UniversityUppsalaSweden
  2. 2.Department of StatisticsUniversity of LundLundSweden

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