## Sub-Bernoulli functions, moment inequalities and strong laws for nonnegative and symmetrized $$U$$-statistics.(English)Zbl 0951.60028

This important article contains the solution to the law of large numbers for $$U$$-statistics of degree 2 based on symmetrized kernels, namely for $U_n:= {1\over n^2}\sum_{i\neq j\leq n}\varepsilon_i \varepsilon_j f(X_i,X_j)\to 0 \text{ a.s.} \tag{1}$ where $$\{\varepsilon_i\}$$ are i.i.d. Rademacher independent of the i.i.d. sequence $$\{X_i\}$$, and $$f$$ is measurable. This problem is difficult and there are several previous papers devoted to it. It is classical that $$E|f|<\infty$$ is sufficient, but it is not necessary. The necessary and sufficient conditions for (1) are given in terms of the distribution of $$f(X_1, X_2)$$ in a relatively complicated way. This is the main result, but the paper also contains nasc’s for normings different from $$n^2$$ as well as the equivalence between convergence of sums and maxima (with the given normalizations), decoupled or not, for $$U$$-statistics of any order $$m$$. And also for multisample $$U$$-statistics. The method of proof is based on estimation of moments and tail probabilities, in particular by an elaborate comparison with sums of products of Bernoulli random variables (sub-Bernoulli functions). Recently, Latała and Zinn [Ann. Probab. (to appear)] have extended the LLN in this paper for symmetrized $$U$$-statistics of order 2 to $$U$$-statistics of any order.
Reviewer: E.Gine (Storrs)

### MSC:

 60F15 Strong limit theorems 60G50 Sums of independent random variables; random walks
Full Text:

### References:

  Arcones, M. and Giné, E. (1993). Limit theorems for U-processes. Ann. Probab. 21 1494-1542. · Zbl 0789.60031  Chow, Y. S. and Teicher, H. (1988). Probability Theory. Springer, New York. · Zbl 0652.60001  Cuzick, J., Giné, 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. · Zbl 0833.60030  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. · Zbl 0827.60014  Giné, 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. Birkhäuser, Boston. · Zbl 0851.60029  Giné, E. and Zinn, J. (1994). A remark on convergence in distribution of U-statistics. Ann. Probab. 22 117-125. · Zbl 0801.60015  Gleser. L. J. (1975). On the distribution of the number of successes in independent trials. Ann. Probab. 3 182-188. · Zbl 0301.60010  Hoeffding, W. (1961). The strong law of large numbers for U-statistics. Institute of Statistics Mimeo Ser. 302, Univ. North Carolina, Chapel Hill. · Zbl 0211.20605  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. · Zbl 0895.60018  Klass, M. J. and Zhang, C.-H. (1994). On the almost sure minimal growth rate of partial sum maxima. Ann. Probab. 22 1857-1878. · Zbl 0857.60023  Ledoux, M. and Talagrand, M. (1991). Probability in Banach spaces. Springer, New York. · Zbl 0748.60004  Serfling, R. J. (1980). Approximation Theorems of Mathematical Statistics. Wiley, New York. · Zbl 0538.62002  Sen, P. K. (1974). On Lp-convergence of U-statistics. Ann. Inst. Statist. Math. 26 55-60. · Zbl 0337.62013  Teicher, H. (1992). Convergence of self-normalized generalized U-statistics. J. Theoret. Probab. 5 391-405. · Zbl 0758.60028  Zhang, C.-H. (1996). Strong laws of large numbers for sums of products. Ann. Probab. 24 1589- 1615. · Zbl 0868.60024
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.