## Abstract

Using the theorem of Fréchet and Kolmogorov about compact subsets of the space L_{p}[0, 1] and Prohorov's theorem about the convergence of measures defined on a complete metric space we proved in [2] the following theorem: Let ξ_{1}(t),…, ξ_{n},(/)>… and ξ (t) be measurable random processes (0 ≤ t ≤ 1) and suppose that there exist numbers C and p ≥ 1 such that E|ξ_{n}(t)^{p}| ≤ C for all n and t. If E|ξ_{n}(t)^{p}|→ £|£(0lp for all /and if for any finite set {t_{1},…, t_{k}] c [0, 1] the joint distribution of £„(f,),…, £„(tk) converges to the joint distribution of ξ (t_{n}),…, ξ (t_{k}), then the distribution of F(ξ_{n}) converges to the distribution of F(ξ) for any continuous functional F on L_{P},[0, 1]. In this paper this theorem is generalized to random measurable functions. The results of the present paper are related to the results of [1], [3], [4].

Original language | English |
---|---|

Pages (from-to) | 322-325 |

Number of pages | 4 |

Journal | Proceedings of the American Mathematical Society |

Volume | 74 |

Issue number | 2 |

DOIs | |

State | Published - May 1979 |

Externally published | Yes |

## Keywords

- Convergence of joint distributions
- Convergence of measures
- Measurable random functions