## The law of large numbers for $$U$$-statistics under absolute regularity.(English)Zbl 0901.60015

The law of large numbers for $$U$$-statistics whose underlying sequence of random variables satisfies a $$\beta$$-mixing condition is considered. The following theorem is proved: Let $$\{X_i\}_{i=1}^{\infty}$$ be a strictly stationary sequence of random variables with values in a measurable space $$S$$. Let $$h:S^m \to R$$ be a symmetric function. Suppose that at least one of the following conditions is satisfied:
(i) For some $$\delta > 2$$, $\sup_{1\leq i_1 < \cdots <i_m<\infty } E[ h(X_{i_1}, \cdots ,X_{i_m})| ^{\delta}]<\infty \quad\text{and}\quad \beta_n \to 0.$ (ii) For some $$0<\delta \leq 1$$ and some $$r> 2\delta^{-1}$$, $\sup_{1\leq i_1 < \cdots <i_m<\infty } E[ h(X_{i_1}, \cdots ,X_{i_m})| ^{1+\delta}]<\infty \quad\text{and}\quad \beta_n = O((\log n)^{-r}).$ (iii) For some $$0< \delta \leq 1$$ and some $$r>0$$, $\sup_{1\leq i_1 < \cdots <i_m<\infty } E[ h(X_{i_1}, \cdots ,X_{i_m})| (\log| h(X_{i_1}, \cdots ,X_{i_m}|)^{1+\delta}]<\infty \quad\text{and}\quad \beta_n =O(n^{-r}).$ Then, $n^{-m} \sum_{1\leq i_1 < \cdots <i_m<n} (h(X_{i_1}, \cdots ,X_{i_m}) - E[h(X_{i_1}, \cdots ,X_{i_m})]) \to 0 \quad \text{a.s.}$ It is mentioned that the conditions in the theorem are very close to being optimal.

### MSC:

 60F15 Strong limit theorems
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