ON OCCUPATION TIMES FOR MARKOFF PROCESSES
ON OCCUPATION TIMES FOR MARKOFF PROCESSES
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DOI：
10.1090/s0002-9947-1957-0084222-7
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发表时间：
1957-02
影响因子：
1.3
通讯作者：
D. Darling;M. Kac
中科院分区：
文献类型：
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作者：
D. Darling;M. Kac
其中u（t）是合适的归一化。如果V（x）是一个集合的特征函数，则ftaV（x（r））dT是该集合的占用时间。主要的结果是，在适当的（但相当普遍的）条件下，极限分布必须是（适当指数的）米塔格-莱弗勒分布。证明方法同样适用于马尔可夫链，特别是独立同分布随机变量的总和。从而得到了Feller [1]，Chung和Kac [2]以及Kallianpur和Robbins [3; 4]的结果的相当大的推广和统一.可以看出，有些冗长的计算，这些作者可以免除凭借基本陶伯定理的卡拉马塔。最后，同分布随机变量部分和序列中符号变化数的分布也将作为我们一般理论的应用而出现。2.一个特殊的cese。为了说明这种方法，并清楚地说明证明一般定理所依据的假设的作用，我们将首先考虑一种特殊情况。设x（t），t^O是二维布朗运动，x（0）=0，V（x）是具有非零Lebesgue测度的有界平面集B的特征函数.让我们计算f0 V（x（T））dr的矩。例如，考虑二阶矩
where u(t) is a suitable normalization. If V(x) is the characteristic function of a set, ftaV(x(r))dT is the occupation time of the set. The principal result is that under suitable (but quite general) conditions the limiting distribution must be the Mittag-Leffler distribution (of an appropriate index). The method of proof is equally applicable to Markoff chains and, in particular, to sums of independent, identically distributed random variables. We thus obtain a considerable generalization and unification of previous results of Feller [l], Chung and Kac [2] and Kallianpur and Robbins [3; 4]. It will be seen that the somewhat lengthy computations of these authors can be dispensed with by virtue of the elementary Tauberian theorem of Karamata. Finally the distribution of the number of changes of sign in a sequence of partial sums of identically distributed random variables will also emerge as an application of our general theory. 2. A special cese. In order to illustrate the method and bring out clearly the role of assumptions under which the general theorem will be proved we shall first consider a special case. Let x(t), t^O be the two dimensional Brownian motion, x(0) =0, and let V(x) he the characteristic function of a bounded plane set B of nonzero Lebesgue measure. Let us calculate the moments of f0V(x(T))dr. Consider e.g. the second moment