In the theory of random fractal, there are two important classes of random sets, one is the class of fractals generated by the paths of stochastic processes and another one is the class of factals generated by statist...In the theory of random fractal, there are two important classes of random sets, one is the class of fractals generated by the paths of stochastic processes and another one is the class of factals generated by statistical contraction operators. Now we will introduce some things about the probability basis and fractal properties of fractals in the last class. The probability basis contains (1) the convergence and measurability of a random recursive setK(ω) as a random element, (2) martingals property. The fractal properties include (3) the character of various similarity, (4) the separability property, (5) the support and zero-one law of distributionP k =P·K ?1, (6) the Hausdorff dimension and Hausdorff exact measure function.展开更多
I.i.d. random sequence is the simplest but very basic one in stochastic processes, and statistically self-similar set is the simplest but very basic one in random recursive sets in the theory of random fractal. Is the...I.i.d. random sequence is the simplest but very basic one in stochastic processes, and statistically self-similar set is the simplest but very basic one in random recursive sets in the theory of random fractal. Is there any relation between i.i.d. random sequence and statistically self-similar set? This paper gives a basic theorem which tells us that the random recursive set generated by a collection of i.i.d. statistical contraction operators is always a statistically self-similar set.展开更多
We have constructed a class of random sets by statistical contraction operators in this paper.When the probability space is some special product space and the statistical contraction operators are affine or similar,th...We have constructed a class of random sets by statistical contraction operators in this paper.When the probability space is some special product space and the statistical contraction operators are affine or similar,these statistically recursive sets are investigated by many authors.It will be very convenient for us to study their distributions and dimensions and measures using our model in this paper.展开更多
The main aim of this paper is to make a classification of random sets K m(ω) constructed in theorem 2.1 and theorem 2.1' in . We provide five criterions for the classification. Many kinds of random sets such...The main aim of this paper is to make a classification of random sets K m(ω) constructed in theorem 2.1 and theorem 2.1' in . We provide five criterions for the classification. Many kinds of random sets such as Hawkes constructed in , Graf constructed in and Mauldin constructed in are the special cases of K m(ω) constructed in ,and then these random sets belong to some model respectively according to our classification.展开更多
文摘In the theory of random fractal, there are two important classes of random sets, one is the class of fractals generated by the paths of stochastic processes and another one is the class of factals generated by statistical contraction operators. Now we will introduce some things about the probability basis and fractal properties of fractals in the last class. The probability basis contains (1) the convergence and measurability of a random recursive setK(ω) as a random element, (2) martingals property. The fractal properties include (3) the character of various similarity, (4) the separability property, (5) the support and zero-one law of distributionP k =P·K ?1, (6) the Hausdorff dimension and Hausdorff exact measure function.
基金Project supported by the National Natural Science Foundation of China the Doctoral Progamme Foundation of China and the Foundation of Wuhan University.
文摘I.i.d. random sequence is the simplest but very basic one in stochastic processes, and statistically self-similar set is the simplest but very basic one in random recursive sets in the theory of random fractal. Is there any relation between i.i.d. random sequence and statistically self-similar set? This paper gives a basic theorem which tells us that the random recursive set generated by a collection of i.i.d. statistical contraction operators is always a statistically self-similar set.
文摘We have constructed a class of random sets by statistical contraction operators in this paper.When the probability space is some special product space and the statistical contraction operators are affine or similar,these statistically recursive sets are investigated by many authors.It will be very convenient for us to study their distributions and dimensions and measures using our model in this paper.
文摘The main aim of this paper is to make a classification of random sets K m(ω) constructed in theorem 2.1 and theorem 2.1' in . We provide five criterions for the classification. Many kinds of random sets such as Hawkes constructed in , Graf constructed in and Mauldin constructed in are the special cases of K m(ω) constructed in ,and then these random sets belong to some model respectively according to our classification.
