An extension of a theorem of S. J. Taylor concerning the multiple points of the symmetric stable process
An extension of a theorem of S. J. Taylor concerning the multiple points of the symmetric stable process
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S.J.泰勒关于对称稳定过程多点定理的推广
DOI：
10.1007/bf00535468
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发表时间：
1967
期刊：
影响因子：
--
通讯作者：
B. Fristedt
中科院分区：
文献类型：
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作者：
B. Fristedt
在[3]中TAYzo~考虑了R n 中指数~的对称稳定过程。如果k是一个正整数，使得~gn(k--1)/k，他以概率1证明不存在k重点。如果n~ 2 且n~> n (k--1)/k，他证明以概率1，k 重点的集合E~ 具有tLausdorff 维数k：c--n (k--1)。我们将取消n~2的限制。如果k=1，则该定理是众所周知的([l])。因此，我们将仅限于 n> 2, k> 1, n~ g> n (k--1)/k 的情况，相当于 n= 3, k= 2, 2~~.>~;然而，这里给出的证明可以很容易地推广到其他情况。在[2]中，推测3维布朗运动的双点集具有I-lausdorff维数I。这个结果将作为特例遵循。在[3]中，TAYLO~无法处理n----3的情况，因为双点集与3维空间的大小相比非常小，以至于无法被独立的稳定过程击中。我们将通过将双点集投影到其中一个坐标轴上并用稳定的过程击中该投影来避免这种情况。
In [3] TAYzo~ considers the symmetric stable process of index~ in R n. If k is a positive integer such that~ gn (k--1)/k, he proves that, with probability one, there are no k-multiple points. If n~ 2 and n~> n (k--1)/k, he proves that, with probability one, the set E~ of k-multiple points has tIausdorff dimension k: c--n (k--1). We shall remove the restriction that n~ 2. If k= 1, the theorem is well-known ([l]). Thus we shall confine ourselves to the case where n> 2, k> 1, n~ g> n (k--1)/k which is equivalent to n= 3, k= 2, 2~~.>~; however the proof given here can easily be generalized to the other cases.In [2] it is conjectured that the set of double points of 3-dimensional Brownian motion has I-lausdorff dimension I. This result will follow as a special case. In [3], TAYLO~ was not able to handle the case n----3 since the set of double points is so small compared to the size of 3-dimensional space that it can not be hit by an independent stable process. We shall avoid this by projecting the set of double points on to one of the coordinate axes and hitting this projection with a stable process.