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

1. 在所有无限可分分布的类中，阐明了半自可分分布类及其嵌套子类的结构。我们介绍了一种由一类分布衍生出一类新的极限分布的方法，即通过独立随机变量的归一化部分和的子序列求极限。我们完整地描述了这一过程的一种不动点。与所有自分解分布的绝对连续性相比，我们发现半自分解分布不一定是绝对连续的。我们还发现了一类总是绝对连续的半自分解分布。我们构造了一些非自分解（或非半自分解）分布的例子，它们在低维空间上的投影是自分解（或半自分解）的。这一特性与稳定分布形成鲜明对比。我们发现半自相似过程的边际分布是可自分解的。我们还发现它们的联合分布多次与自分解分布的嵌套子类密切相关。也证明了自分解分布和自相似过程之间的相似观察结果是成立的。我们成功地定义了多元型G分布，并找到了其自分解的充分必要条件。我们还定义了G型分布的一系列嵌套子类，并成功地对无限可分分布类进行了新的细化。

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.