The law of large numbers for \(U\)-statistics under absolute regularity
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.
strictly stationary sequence of random variables
scientific article; zbMATH DE number 1171020