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

Zhang, Cun-Hui

Ann. Probab. 27, No. 1, 432-453 (1999).

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

Keywords:

sub-Bernoulli functions; strong law of large number; distribution inequalities; \(U\)-statistics

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