## The Annals of Probability

### Sub-Bernoulli Functions, Moment Inequalities and Strong Laws for Nonnegative and symmetrized U-Statistics

Cun-Hui Zhang

#### Abstract

This paper concerns moment and tail probability inequalities and the strong law of large numbers for $U$-statistics with nonnegative or symmetrized kernels and their multisample and decoupled versions. Sub-Bernoulli functions are used to obtain the moment and tail probability inequalities, which are then used to obtain necessary and sufficient conditions for the almost sure convergence to zero of normalized $U$-statistics with nonnegative or completely symmetrized kernels, without further regularity conditions on the kernel or the distribution of the population, for normalizing constants satisfying a simple condition. Moments of $U$-statistics are bounded from above and below by that of maxima of certain kernels, up to scaling constants. The multisample and decoupled versions of these results are also considered.

#### Article information

Source
Ann. Probab., Volume 27, Number 1 (1999), 432-453.

Dates
First available in Project Euclid: 29 May 2002

https://projecteuclid.org/euclid.aop/1022677268

Digital Object Identifier
doi:10.1214/aop/1022677268

Mathematical Reviews number (MathSciNet)
MR1681165

Zentralblatt MATH identifier
0951.60028

Subjects
Primary: 60F15: Strong theorems
Secondary: 60G50: Sums of independent random variables; random walks

#### Citation

Zhang, Cun-Hui. Sub-Bernoulli Functions, Moment Inequalities and Strong Laws for Nonnegative and symmetrized U -Statistics. Ann. Probab. 27 (1999), no. 1, 432--453. doi:10.1214/aop/1022677268. https://projecteuclid.org/euclid.aop/1022677268

#### References

• Arcones, M. and Gin´e, E. (1993). Limit theorems for U-processes. Ann. Probab. 21 1494-1542.
• Chow, Y. S. and Teicher, H. (1988). Probability Theory. Springer, New York.
• Cuzick, J., Gin´e, E. and Zinn, J. (1995). Laws of large numbers for quadratic forms, maxima of products and truncated sums of i.i.d. random variables. Ann. Probab. 23 292-333.
• de la Pe na, V. and Montgomery-Smith, S. J. (1995). Decoupling inequalities for the tail probabilities of multivariate U-statistics. Ann. Probab. 23 806-816.
• Gin´e, E. and Zinn, J. (1992). Marcinkiewicz type laws of large numbers and convergence of moments for U-statistics. In Probability in Banach Spaces 8 (R. M. Dudley, M. G. Hahn and J. Kuelbs, eds.) 273-291. Birkh¨auser, Boston.
• Gin´e, E. and Zinn, J. (1994). A remark on convergence in distribution of U-statistics. Ann. Probab. 22 117-125.
• Gleser. L. J. (1975). On the distribution of the number of successes in independent trials. Ann. Probab. 3 182-188.
• Hoeffding, W. (1961). The strong law of large numbers for U-statistics. Institute of Statistics Mimeo Ser. 302, Univ. North Carolina, Chapel Hill.
• Klass, M. J. and Nowicki (1997). Order of magnitude bounds for expectations of 2-functions of non-negative random bilinear forms and generalized U-statistics. Ann. Probab. 25 1471-1501.
• Klass, M. J. and Zhang, C.-H. (1994). On the almost sure minimal growth rate of partial sum maxima. Ann. Probab. 22 1857-1878.
• Ledoux, M. and Talagrand, M. (1991). Probability in Banach spaces. Springer, New York.
• Serfling, R. J. (1980). Approximation Theorems of Mathematical Statistics. Wiley, New York.
• Sen, P. K. (1974). On Lp-convergence of U-statistics. Ann. Inst. Statist. Math. 26 55-60.
• Teicher, H. (1992). Convergence of self-normalized generalized U-statistics. J. Theoret. Probab. 5 391-405.
• Zhang, C.-H. (1996). Strong laws of large numbers for sums of products. Ann. Probab. 24 1589- 1615.