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On the dissipation of partial sums from a stationary strongly mixing sequence. (English) Zbl 0815.60031

The author investigates the distribution of partial sums \(S_ n = \sum^ n_{k = 1} X_ k\) of strictly stationary random variables. For strongly mixing sequences the following dichotomy is established: The sequence \((S_ n)^ \infty_{n = 1}\) is either (i) shift-tight, i.e. \(S_ n' = S_ n - \text{median}(S_ n)\) is tight, or (ii) completely dissipated, i.e. \(\lim_{n \to \infty} \sup_{a \in \mathbb{R}} P(S_ n \in [a,a + C]) = 0\) for all \(C > 0\). In a second theorem the author proves that for strongly mixing sequences whose maximal correlation coefficient is strictly less than 1, only (ii) can hold.

MSC:

60G10 Stationary stochastic processes
60F05 Central limit and other weak theorems
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