CONVERGENCE ON RANDOMLY TRIMMED SUMS WITH A DEPENDENT SAMPLE

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CONVERGENCE ON RANDOMLY TRIMMED SUMS WITH A DEPENDENT SAMPLE

Let $ \{X_n\} $ be a sequence of random variables and $ X_{n1}\le{X_{n2}}\le\cdots \le{X_{nn}} $ their order statistics. In this paper a central limit theorem and a strong law of large numbers for randomly trimmed sums $T_n =\sum_{i=\alpha_n+1}^{\beta_n}\limits X_{ni}$ are established in the case that $\alpha_n$ and $\beta_n$ are positive integer-valued random variables such that $\alpha_n/n$ and $\beta_n/n$ converge to random variables $\alpha$ and $\beta$ respectively with $0\le\alpha<\beta\le1$ in certain sense, and $\{X_n\}$ is a $\varphi$-mixing sequence.