A strong law of large numbers for identically distributed vector lattice- valued random variables. (English) Zbl 0599.60038

For over twenty years many authors have devoted their attention to the laws of large numbers and to similar convergence theorems, i.e. the results on the convergence of weighted sums of random variables in one or another sense. While a number of interesting results has been produced for the norm topology, the theory is much less developed for the weak topology and almost completely neglected is the convergence with respect to the order. It is the latter case I intend to discuss. The main reason for doing so is that in a number of spaces the order convergence is stronger than the topological one (e.g. \(L^ p\)-spaces, \(1\leq p<\infty)\).


60F25 \(L^p\)-limit theorems
60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
60F15 Strong limit theorems
Full Text: EuDML


[1] PADGETT W. J., TAYLOR R. L.: Laws of Large Numbers for Normed Linear Spaces and Certain Fréchet Spaces. Springer, Berlin 1973. · Zbl 0275.60010 · doi:10.1007/BFb0069731
[2] POTOCKÝ R.: Laws of large numbers in Banach lattices. Acta F.R.N. Univ, Comen.) · Zbl 0538.60010
[3] LUXEMBURG W. A., ZAANEN C. C.: Riesz Spaces I. Amsterdam 1971. · Zbl 0231.46014
[4] SCHAFFER H. H.: Banach Lattices and Positive Operators. Berlin 1974.
[5] FREMLIN D. H.: Topological Riesz Spaces and Measure Theory. Cambridge 1974. · Zbl 0273.46035
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.