On the strong law for arrays and for the bootstrap mean and variance. (English) Zbl 0883.60024

Hu, Tien-Chung; Taylor, R. L.

Int. J. Math. Math. Sci. 20, No. 2, 375-382 (1997).

The strong law of large numbers for a triangular array \(\{X_{ni}\), \(1\leq i \leq n, n\geq 1\}\) of row-wise independent (but neither necessarily identically distributed nor independent between rows) random variables is established under conditions similar to those of Chung. Further this result is related to verifying a known fact of consistency of the bootstrap mean and bootstrap variance [cf. S. C\"sorgö, Stat. Probab. Lett. 14, No. 1, 1-7 (1992; Zbl 0752.62034)]. The authors present a new, fairly different approach to this problem in a natural formulation. Let us mention that in Theorem 2.1 not only a.s. but even complete convergence can be stated.

Reviewer: A.I.Volodin (Kazan’)

MSC:

60F15	Strong limit theorems
60B12	Limit theorems for vector-valued random variables (infinite-dimensional case)

Keywords:

strong law of large numbers; row-wise independent; triangular arrays; bootstrap mean and variance; Chung’s strong law of large numbers

Citations:

Zbl 0752.62034