On the strong law of large numbers for pairwise independent random variables. (English) Zbl 0534.60028

Csörgő, S.; Tandori, K.; Totík, V.

Acta Math. Hung. 42, 319-330 (1983).

Using a method of N. Etemadi [Z. Wahrscheinlichkeitstheor. Verw. Geb. 55, 119-122 (1981; Zbl 0438.60027)] the authors prove: if \(X_ 1,X_ 2,..\). are pairwise independent r.v.’s with \[ \sum^{\infty}_{m=1}m^{-2}Var(X_ m) < \infty \tag{\text{i}} \] and \[ n^{-1}\sum^{n}_{m=1}E| X_ m-EX_ m| = O(1) \tag{\text{ii}} \] then the strong law of large numbers holds i.e. \[ \lim n^{-1} \sum^{n}_{m=1} (X_ m-EX_ m) = 0 \quad\text{a.s.} \] It is also proved that the theorem does not hold true if the condition of pairwise independence is replaced by orthogonality. The necessity of condition (ii) is also investigated.

Reviewer: P.Révész

MSC:

60F15	Strong limit theorems
40A30	Convergence and divergence of series and sequences of functions

Keywords:

law of large numbers; pairwise independence; orthogonal random variables

References:

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