TY - JOUR
T1 - Convergence of measurable random functions
AU - Grinblat, L.
PY - 1979/5
Y1 - 1979/5
N2 - Using the theorem of Fréchet and Kolmogorov about compact subsets of the space Lp[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 ξ1(t),…, ξn,(/)>… and ξ (t) be measurable random processes (0 ≤ t ≤ 1) and suppose that there exist numbers C and p ≥ 1 such that E|ξn(t)p| ≤ C for all n and t. If E|ξn(t)p|→ £|£(0lp for all /and if for any finite set {t1,…, tk] c [0, 1] the joint distribution of £„(f,),…, £„(tk) converges to the joint distribution of ξ (tn),…, ξ (tk), then the distribution of F(ξn) converges to the distribution of F(ξ) for any continuous functional F on LP,[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].
AB - Using the theorem of Fréchet and Kolmogorov about compact subsets of the space Lp[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 ξ1(t),…, ξn,(/)>… and ξ (t) be measurable random processes (0 ≤ t ≤ 1) and suppose that there exist numbers C and p ≥ 1 such that E|ξn(t)p| ≤ C for all n and t. If E|ξn(t)p|→ £|£(0lp for all /and if for any finite set {t1,…, tk] c [0, 1] the joint distribution of £„(f,),…, £„(tk) converges to the joint distribution of ξ (tn),…, ξ (tk), then the distribution of F(ξn) converges to the distribution of F(ξ) for any continuous functional F on LP,[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].
KW - Convergence of joint distributions
KW - Convergence of measures
KW - Measurable random functions
UR - http://www.scopus.com/inward/record.url?scp=84966202196&partnerID=8YFLogxK
U2 - 10.1090/S0002-9939-1979-0524310-1
DO - 10.1090/S0002-9939-1979-0524310-1
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AN - SCOPUS:84966202196
SN - 0002-9939
VL - 74
SP - 322
EP - 325
JO - Proceedings of the American Mathematical Society
JF - Proceedings of the American Mathematical Society
IS - 2
ER -