Convergence on randomly trimmed sums with a \(\varphi\)-mixing sample (Q674655)

Let \(\{X_n\), \(n\geq 1\}\) be a sequence of random variables with a common distribution function, and for every \(n \geq 1\) let \(X_{n1} \leq X_{n2} \leq \ldots \leq X_{nn}\) be the order statistics of \(X_1, X_2,\ldots, X_n\). The paper presents a strong law of large numbers and a central limit theorem for the trimmed sums \(\sum_{i=\alpha_n+1}^{\beta_n} X_{ni}\) if the trimming levels \(0 \leq\alpha_n < \beta_n \leq n\) are integer-valued random variables and if the underlying data sequence \(\{ X_n, n \geq 1\}\) is \(\varphi\)-mixing. No proofs are given.