On the optimality of McLeish’s conditions for the central limit theorem. (Sur l’optimalité des conditions de McLeish pour le théorème limite central.) (English. French summary) Zbl 1321.60037
In the main result of this paper, the author gives an explicit construction of a family of stationary ergodic sequences \((X_i)_{i\in\mathbb{N}}\) of square integrable, centred random variables whose sum \(S_n=X_1+\dots+X_n\) has variance which is (asymptotically) linear but for which the central limit theorem does not hold, since \(n^{-1/2}S_n\) fails to converge in distribution. The construction employed by the author shows that the conditions for the central limit theorem due to D. L. McLeish [Z. Wahrscheinlichkeitstheor. Verw. Geb. 32, 165–178 (1975; Zbl 0288.60034)] cannot be relaxed in general.
Reviewer: Fraser Daly (Edinburgh)
MSC:
| 60F05 | Central limit and other weak theorems |
Citations:
Zbl 0288.60034References:
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