On convergence in variation of the distributions of multivariate point processes
On convergence in variation of the distributions of multivariate point processes
复制标题
论多元点过程分布变化的收敛性
DOI：
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发表时间：
1983
期刊：
影响因子：
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通讯作者：
R. Liptser
中科院分区：
文献类型：
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作者：
Y. Kabanov;R. Liptser
本文的目的是建立多元点过程的补偿器给出的分布按变差收敛的充要条件。主要结果如下：如果极限补偿器是有限的，则变差收敛成立的充要条件是补偿器之间的差值的变化在概率上趋于零。定理1给出了精确的表述，它推广了[6]中的定理6和定理7，其中在假设极限过程有一个非随机补偿器的情况下考虑了计数过程的情况，因此是一个具有独立增量的过程。定理1的A)部分的证明是基于给出补偿器之间的变化距离的上界的重要的不等式(定理2)。定理2的证明技巧源于[-8]中定理2的证明，并且主要基于密度过程的结构和性质。(有趣的是，与定理1相反，关于计数过程的有限维分布关于其补偿器的弱收敛的充分条件(见[1，8])不是必要条件。E.L.普雷斯曼和I.M.索宁向我们传达了一个反例。)作为定理2应用的一个例子，我们给出了经验分布函数收敛到Poisson过程的速度的估计。论文的结构如下。在第二节中，我们给出了主要结果的陈述。教派。定理3和4包含定理2和1的证明。第五节致力于定理1和定理2的一些推广。在第六节中我们给出一个例子。
The aim of the present paper is to establish necessary and sufficient condition for convergence in variation of the distributions of multivariate point processes given by their compensators. The main result is the following: if the limiting compensator is finite then the convergence in variation holds if and only if the variations of the differences between compensators tends to zero "in probability". The precise statement is given in Theorem 1. It extends Theorems 6 and 7 of [6] where the case of counting processes has been considered under assumption that the limiting process has a nonrandom compensator, and is therefore a process with independent increments. The proof of the part A) of Theorem 1 is based on important inequalities which give upper bounds for the variation distance between compensators (Theorem 2). The technique used in the proof of Theorem 2 originates from the proof of Theorem 2 in [-8] and based heavily on the structure and properties of a density process. (It is interesting to note that, in contrast with Theorem 1, sufficient conditions for weak convergence of finite-dimensional distributions for counting processes in terms of their compensators (see [1,8]) are not necessary. There is a counter example communicated to us by E.L. Presman and I.M. Sonin.) As an example of an application of Theorem 2, we give an estimate of the rate of convergence of an empirical distribution function to a Poisson process. The structure of the paper is the following. In Sect. 2 we give the statement of principle results. The Sects. 3 and 4 contain the proofs of Theorem 2 and 1. The Sect. 5 is devoted to some generalizations of Theorems 1 and 2. In Sect. 6 we give an example.