Abstract
This chapter discusses sub-sequences of sequences of random variables. A sub-sequence of a stationary sequence is, in general, nonstationary, except when the original sequence is a sequence of symmetrically dependent random variables. Any sub-sequence of a sequence of symmetrically dependent random variables is a stationary sequence. Conversely, if any sub-sequence of a stationary sequence is stationary, then the original sequence is a sequence of symmetrically dependent random variables. The investigation of a random sub-sequence of a strongly stationary sequence is very easy. Any sequence of random variables, bounded in a certain sense, contains a sub-sequence obeying the law of large numbers and the law of the iterated logarithm.
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