Integrability of expected increments of point processes and a related random change of scale
Integrability of expected increments of point processes and a related random change of scale
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点过程的预期增量和相关的尺度随机变化的可积性
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发表时间：
1972
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通讯作者：
F. Papangelou
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作者：
F. Papangelou
给定实线R上具有有限强度的平稳点过程，用N(Q) （R中的Q Borel集合）表示该过程在Q中抛出的随机点数，用^ （t s R）表示（- co, t）中发生的事件的c场。主要结果如下。如果对于区间[b， c]的每个分区A ={¿»=f0<fi<••••<fn +i = c}，我们设SA（co) = IJ.0 E(N[ÍV, f,+i)|3»，），则lim4 5Ä（cu)= W(A,, [b, c)）在maxos，än (fv +i - fv) ->-0 （as收敛需要明智地选择版本）存在于平均值中。如果[0,0]的随机变换/»W（<a,[0, /))对自身是a.s.连续的（即无跳跃），则它将过程的非负点转化为速率为1且与^0无关的泊松过程，比值c~1E(N[0, e)\^0）收敛于e.|00。其收敛于均值（以及函数W[Q， t]在（0，»）上的绝对连续性）的一个充分必要条件是Palm条件概率P0相对于cr-场&<，上的绝对概率P的绝对连续性。进一步的结果见§1。
Given a stationary point process with finite intensity on the real line R, denote by N(Q) (Q Borel set in R) the random number of points that the process throws in Q and by ^ (t s R) the c-field of events that happen in ( — co, t). The main results are the following. If for each partition A = {¿»=f0<fi< • • • <fn + i = c} of an interval [b, c] we set SA(co) = IJ.0 E(N[ÍV, f,+i)|3»,) then lim4 5Ä(cu)= W(a,, [b, c)) exists a.s. and in the mean when maxos,än (fv + i — fv) ->-0 (the a.s. convergence requires a judicious choice of versions). If the random transformation / » W(<a, [0, /)) of [0, oo) onto itself is a.s. continuous (i.e. without jumps), then it transforms the nonnegative points of the process into a Poisson process with rate 1 and independent of ^oThe ratio c~1E(N[0, e)\^0) converges a.s. as e|0. A necessary and sufficient condition for its convergence in the mean (as well as for the a.s. absolute continuity of the function W[Q, t ) on (0, »)) is the absolute continuity of the Palm conditional probability P0 relative to the absolute probability P on the cr-field &<,. Further results are described in §1.