Statistically self-similar sets and random invariant measures. (Chinese. English summary) Zbl 1009.60043

Yu, Jinghu; Yan, Yunqi

Acta Math. Sci., Ser. A, Chin. Ed. 22, No. 4, 471-476 (2002).

Summary: Let \(X\) be a complete separable metric space and \(K(\omega)\) be the statistically self-similar sets defined by S. Graf [Probab. Theory Relat. Fields 74, 357-392 (1987; Zbl 0591.60005)]. We construct a series of random invariant measures \({\mathcal U}^*_n\) \((n\geq 1)\), which are generalizations of invariant measures studied by J. E. Hutchinson [Indiana Univ. Math. J. 30, 713-747 (1981; Zbl 0598.28011)] and prove that there exists a probability measure \({\mathcal U}^*\) with \(\text{supp }{\mathcal U}^*= K(\omega)\) such that \({\mathcal U}^*_n\to{\mathcal U}^*\) (weakly); finally, we obtain some local properties of \({\mathcal U}^*_n\).

MSC:

60G57	Random measures
28A80	Fractals

Keywords:

statistically self-similar set; random invariant measure; weak convergence; local properties