Convergence on randomly trimmed sums with a dependent sample

Let {X-n} be a sequence of random variables and X-n1 less than or equal to X-n2 less than or equal to ... less than or equal to 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 = (i=alpha n+1)Sigma(beta n) 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 less than or equal to alpha < beta less than or equal to 1 in certain sense, and {X-n} is a phi-mixing sequenc.