Multiple points for the sample paths of the symmetric stable process
Multiple points for the sample paths of the symmetric stable process
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DOI：
10.1007/bf00533062
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发表时间：
1966-09
期刊：
影响因子：
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通讯作者：
Stephen Taylor;Stephen Taylor
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文献类型：
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作者：
Stephen Taylor;Stephen Taylor
许多作者研究了欧氏n-空间Rn中~，0<~=< 2阶对称稳定过程的样本路的性质（例见[1，2，13，16]）。众所周知，对于1<~ 2，n= 1，该过程是点递归的，因此事实上对于任何给定的x~ R1，x被访问c次的概率为1（c是连续统的基数），因此不存在多点的存在问题。对于其他值的~，n的过程不是点递归的，但可能有一些点的路径访问一次以上。一个至少被访问两次的点被称为双点，而如果它在不少于k个不同的时刻被访问，我们称之为/c-多点。本文完全解决了0<~< 2时G-重点的存在性问题，并进一步考虑了G-重点集在非空条件下的扩张问题。其中~= 2的情形对应于布朗运动，并已在文献[6- 9]中完全解决，主要是由于DVORETZKY，ERD 0 S和KAKUTANI.对于f c= 2和g的其他值，TAKEUCKI [16]已经证明，如果2~~>，则双点存在的概率为1。89，但对于0< g G 89，路径以概率1进入任何点两次。我们将利用[16]中的估计，尽管我们的方法必须有所不同。
Properties of the sample path for the symmetric stable process of order~, 0<~=< 2 in Euclidean n-space, Rn have been studied by many authors (see for example [1, 2, 13, 16]). It is well known that for 1<~~ 2, n= 1, the process is point recurrent so that in fact for any given x~ R 1 there is probability 1 that x will be visited c times (c is the cardinal of the continuum) and there is therefore no problem about the existence of multiple points. For other values of~, n the process is not point recurrent but there may be some points which the path visits more than once. A point visited at least twice is called a double point, while if it is visited at not less than k different time instants we call it a/c-multiple point. In the present paper we completely settle the question of the existence of/c-multiple points for 0<~< 2 and go on to consider the extent of the set of Gmultiple points when this is known to be non-void. The case~= 2 corresponds to Brownian motion and has previously been completely settled in the series of papers [6--9] mainly due to DVORETZKY, ERD0S and KAKUTANI. For/c= 2 and other values of g, TAKEUCKI [16] has shown that double points exist with probability 1 if 2~~.> 89 but that for 0< g G 89 the path enters no point twice with probability 1. We will make use of the estimates in [16], though our method has to be somewhat different.