The Annals of Probability

Optimal Bounds in Non-Gaussian Limit Theorems for U-Statistics

V. Bentkus and F. Götze

Full-text: Open access


Let $X,X_1,X_2,\ldots$ be i.i.d. random variables taking values in a measurable space $\mathscr{X}$. Let $\phi(x,y)$ and $\phi_1(x)$ denote measurable functions of the arguments $x,y\in\mathscr{X}$. Assuming that the kernel $\phi$ is symmetric and the $\mathbf{E}\phi(x,X)= 0$, for all $x$, and $\mathbf{E}\phi_1(X) = 0$, we consider $U$-statistics of type

\[ T = N^{-1} \textstyle\sum\limits_{1\le j < k\le N} \phi(X_j,X_k) + N^{-1/2} \textstyle\sum\limits_{1\le j\le N}\phi_1(X_j). \]


Is is known that the conditions $\mathbf{E}\phi^2(X,X_1)<\infty$ and $\mathbf{E}\phi_1^2(X)<\infty$ imply that the distribution function of $T$, say $F$, has a limit, say $F_0$, which can be described in terms of the eigenvalues of the Hilbert-Schmidt operator associated with the kernel $\phi(x,y)$. Under optimal moment conditions, we prove that


\[ \Delta_N = \sup_x |F(x) - F_0(x) - F_1(x)|= \mathcal{O}(N^{-1}), \]

provided that at least nine eigenvalues of the operator do not vanish. Here $F_1$ denotes an Edgeworth-type correction. We provide explicit bounds for $\Delta_N$ and for the concentration functions of statistics of type $T$.

Article information

Ann. Probab., Volume 27, Number 1 (1999), 454-521.

First available in Project Euclid: 29 May 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62E20: Asymptotic distribution theory
Secondary: 60F05: Central limit and other weak theorems

$U$-statistics degenerate $U$-statistics von Mises statistics symmetric statistics central limit theorem convergence rates Berry-Esseen bounds Edgeworth expansions second order efficiency


Bentkus, V.; Götze, F. Optimal Bounds in Non-Gaussian Limit Theorems for U -Statistics. Ann. Probab. 27 (1999), no. 1, 454--521. doi:10.1214/aop/1022677269.

Export citation


  • ANDERSON, T. W. and DARLING, D. A. 1952. Asymptotic theory of certain ``goodness of fit'' criteria based on stochastic processes. Ann. Math. Statist. 23 193 212.
  • BENTKUS, V. 1984. Asymptotic expansions for distributions of sums of independent random elements of a Hilbert space. Lithuanian Math. J. 24 305 319.
  • BENTKUS, V. and GOTZE, F. 1995. On the lattice point problem for ellipsoids. Russian Acad. Sci. ¨ Dokl. Mach. 343 439 440.
  • BENTKUS, V. and GOTZE, F. 1996. Optimal rates of convergence in the CLT for quadratic forms. ¨ Ann. Probab. 24 466 490.
  • BENTKUS, V. and GOTZE, F. 1997a. On the lattice point problem for ellipsoids. Acta Arith. 80 ¨ 101 125.
  • BENTKUS, V. and GOTZE, F. 1997b. Uniform rates of convergence in the CLT for quadratic forms ¨ in multidimensional spaces. Prob. Theory and Related Fields 109 367 416.
  • BENTKUS, V. and GOTZE, F. 1997c. Optimal bounds in non-Gaussian limit theorems for U¨ Z statistics. Preprint 97-077 SFB 343, Univ. Bielefeld http:// www.mathematik.. sfb343 / preprints.html.
  • BENTKUS, V. and GOTZE, F. 1997d. Lattice point problems and distribution of values of ¨ Z quadratic forms. Preprint 96-111 SFB 343, Univ. Bielefeld http:// www.mathemat. sfb343 / preprints.html.
  • BENTKUS, V., GOTZE, F. and PAULAUSKAS, V. 1996. Bounds for the accuracy of Poissonian ¨ approximations of stable laws. Stochastic Process. Appl. 65 55 58.
  • BENTKUS, V., GOTZE, F. and ZAITSEV, A. YU. 1997. Approximation of distributions of quadratic ¨ forms of independent random vectors by accompanying laws. Theory Probab. Appl. 42 308 335.
  • BENTKUS, V., GOTZE, F. and ZITIKIS, R. 1993. Asymptotic expansions in the integral and local ¨ limit theorems in Banach spaces with applications to -statistics. J. Theoret. Probab. 6 727 780.
  • CHUNG, K. L. 1974. A course in probability theory. Academic Press, New York.
  • BORELL, C. 1974. Convex measure on locally convex spaces. Arkiv. Mat. 12 239 252.2
  • CSORGO, S. 1976. On an asymptotic expansion for the von Mises -statistics. Acta. Sci. Math. ¨ ¨ 38 45 67.
  • ESSEEN, C.-G. 1945. Fourier analysis of distribution functions. Acta. Math. 77 1 125.
  • GOTZE, F. 1979. Asymptotic expansion for bivariate von Mises functionals.Wahrsch. Verw. ¨ Gebiete 50 333 355. Z.
  • GOTZE, F. 1984. Expansions for von Mises functions.Wahrsch. Verw. Gebiete 65 599 625. ¨ Z.
  • GOTZE, F. and ZITIKIS, R. 1995. Edgeworth expansions and bootstrap for degenerate von Mises ¨ statistics. Probab. Math. Statist. 15 327 351.Z. Z.
  • HARDY, G. H. 1916. The average order of the arithmetical functions P x and x. Proc. London Math. Soc. 15 192 213.
  • ITO, K. 1951. Multiple Wiener integral. J. Math. Soc. Japan 3 157 169.
  • KANDELAKI, N. P. 1965. On limit theorem in Hilbert space. Trudy Vychisl. Centra Akad. NaukGruzin. SSR 11 46 55 in Russian. Z.
  • KIEFER, J. 1972. Skorohod embedding of multivariate r.v.'s and the sample d.f.Wahrsch. Verw Gebiete 24 1 35.
  • KOROLJUK, V. S. and BOROVSKICH, YU. V. 1994. Theory of U-statistics. Kluwer, Dordrecht.
  • LANDAU, E. 1915. Zur analytischen Zahlentheorie der definiten quadratischen Formen. Sitzber. Preuss. Akad. Wiss. 31 458 476.
  • LEE, A. J. 1990. U-statistics, Theory and practice. Dekker, New York.
  • MAJOR, P. 1981. Multiple Wiener Integral. Lecture Notes in Math. 849 157 169. Springer, New York.
  • NAGAEV, S. V. and CHEBOTAREV, V. I. 1986. A refinement of the error: estimate of the normal approximation in a Hilbert space. Siberian Math. J. 27 434 450.
  • PAULAUSKAS, V. 1976. On the rate of convergence in the central limit theorem in certain Banach spaces. Theory Probab. Appl. 21 754 769.
  • PETROV, V. V. 1975. Sums of Independent Random Variables. Springer, Berlin.
  • SAZONOV, V. V. 1969. An improvement of a convergence rate estimate. Theory Probab. Appl. 14 640 651.
  • SENATOV, V. V. 1989. On the estimate of the rate of convergence in the central limit theorem in Hilbert space. Stability Problems for Stochastic Models. Lecture Notes in Math. 1412 309 327. Springer, Berlin.2 Z.
  • SMIRNOV, N. V. 1937. On the distribution of the criterion of von Mises. Rec. Math. NS 2973 993 in Russian. ¨ Z.
  • WEYL, H. 1915. Uber die Gleichverteilung der Zahlen mod-Eins. Mathem. Ann. 77 313 352.
  • YARNOLD, J. K. 1972. Asymptotic approximations for the probability that a sum of lattice random vectors lies in a convex set. Ann. Math. Statist. 43 1566 1580.
  • YURINSKII, V. V. 1982. On the accuracy of normal approximation of the probability of hitting a ball. Theory Probab. Appl. 27 280 289.
  • YURINSKII, V. V. 1995. Sums and Gaussian Vectors. Lecture Notes in Math. 1617. Springer, Berlin.
  • ZALESSKII, B. A., SAZONOV, V. V. and ULYANOV, V. V. 1988. A sharp estimate for the accuracy of the normal approximation in a Hilbert space. Theory Probab. Appl. 33 700 701.