A study on semi-selfdecomposable distributions and semi-selfsimilar stochastic processes

1.在所有无限可分分布类中，明确了半自分解分布类及其嵌套子类的结构。通过独立随机变量的归一化部分和的一些子序列取极限，我们介绍了一种由一类分布导出的新的极限分布的方法。我们完全描述了这一过程的一种固定点。与所有自可分解分布的绝对连续性相比，我们发现半自可分解分布不一定是绝对连续的。我们还发现了半自可分解分布的一个子类，它们总是绝对连续的。我们构造了一些不可自分解(或非半自可分解)分布的例子，它们到低维空间的投影是自可分解(或半自可分解)的。这一性质与稳定分布形成鲜明对比。我们发现，一次半自相似过程的边际分布是可自分解的。我们还发现，它们在若干时刻的联合分布与自分解分布的嵌套子类密切相关。同时也证明了在可自分解分布和自相似过程之间，类似的观测结果仍然成立。我们成功地定义了多元G型分布，并得到了它们是自分解的充要条件。我们还定义了G型分布的嵌套子类序列，并成功地对无限可分分布类进行了新的精化。

1. The structure of the classes of semi-selfdecomposable distributions and its nested subclasses were clarified among the class of all infinitely divisible distributions. We introduced a way of making a new class of limitinz distributions derived from a class of distributions by taking the limit through some subsequence of normalized partial sums of independent random variables. We characterized completely a sort of a fixed point of this procedure.2. In contrast to the absolute continuity of all selfdecomposable distributions, we found that semi-selfdecomposable distributions are not necessarily absolutely continuous. We also found a subclass of semi-selfdecomposable distributions which are always absolutely continuous.3. We constructed some examples of non-selfdecomposable (or non-semi-selfdecomposable) distributions whose projections to lower dimensional spaces are selfdecomposable (or semi-selfdecomposable). This property has a sharp contrast to stable distributions.4. We found that the marginal distributions of semi-selfsimilar processes at a time is selfdecomposable. We also found that their joint distributions at several times are closely related to nested subclasses of selfdecomposable distributions. It is also proved that similar observations remain true between selfdecomposable distributions and selfsimilar processes.5. We succeeded in defining multivariate type G distributions and found a necessary and sufficient condition for that they are selfdecomposable. We also defined a sequence of nested subclasses of type G distributions and succeeded in making a new refinement of the class of infinitely divisible distributions.