A strong law of large numbers for identically distributed vector lattice- valued random variables.

*(English)*Zbl 0599.60038For 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, \(1\leq p<\infty)\).

##### MSC:

60F25 | \(L^p\)-limit theorems |

60B12 | Limit theorems for vector-valued random variables (infinite-dimensional case) |

60F15 | Strong limit theorems |

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##### References:

[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 |

[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 |

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