Poisson limits for U-statistics

• Published in: Stochastic processes and their applications Vol. 99; no. 1; pp. 137 - 157
• Main Authors: Dabrowski, André R., Dehling, Herold G., Mikosch, Thomas, Sharipov, Olimjon
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01-05-2002; Elsevier Science; Elsevier
• Series: Stochastic Processes and their Applications
• Subjects: Correlation dimension; Exact sciences and technology; Hill estimator; Mathematics; Poisson random measure; Probability and statistics; Probability theory and stochastic processes; Sciences and techniques of general use; Self-normalized sum; Stable distribution; Stein–Chen method; Stochastic processes; Takens estimator; U-statistic; U-statistic Poisson random measure Correlation dimension Takens estimator Hill estimator Stable distribution Stein-Chen method Self-normalized sum; secondary: 60F05; U-statistic; Stein–Chen method; Takens estimator; Correlation dimension; primary: 60G55; 60G70; 62F12; Poisson random measure; Self-normalized sum; Hill estimator; Stable distribution; Kernels; Correlation; Point process; Asymptotic behavior; U statistic; Asymptotic convergence; Poisson process; Variance
• ISSN: 0304-4149; 1879-209X
• DOI: 10.1016/S0304-4149(01)00153-3

We study Poisson limits for
U-statistics with non-negative kernels. The limit theory is derived from the Poisson convergence of suitable point processes of
U-statistics structure. We apply these results to derive infinite variance stable limits for
U-statistics with a regularly varying kernel and to determine the index of regular variation of the left tail of the kernel. The latter is known as correlation dimension. We use the point process convergence to study the asymptotic behavior of some standard estimators of this dimension.