Integrability of expected increments of point processes and a related random change of scale

Abstract:

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 ${\mathcal {F}_t}(t \in R)$ the $\sigma$-field of events that happen in $( - \infty ,t)$. The main results are the following. If for each partition $\Delta = \{ b = {\xi _0} < {\xi _1} < \cdots < {\xi _{n + 1}} = c\}$ of an interval [b, c] we set ${S_\Delta }(\omega ) = \sum \nolimits _{\nu = 0}^n {E(N[{\xi _\nu },{\xi _{\nu + 1}})|{\mathcal {F}_{{\xi _\nu }}})}$ then ${\lim _\Delta }{S_\Delta }(\omega ) = W(\omega ,[b,c))$ exists a.s. and in the mean when ${\max _{0 \leqq \nu \leqq n}}({\xi _{\nu + 1}} - {\xi _\nu }) \to 0$ (the a.s. convergence requires a judicious choice of versions). If the random transformation $t \Rightarrow W(\omega ,[0,1))$ of $[0,\infty )$ 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 ${\mathcal {F}_0}$. The ratio ${\varepsilon ^{ - 1}}E(N[0,\varepsilon )|{\mathcal {F}_0})$ converges a.s. as $\varepsilon \downarrow 0$. A necessary and sufficient condition for its convergence in the mean (as well as for the a.s. absolute continuity of the function $W[0,t)$ on $(0,\infty ))$ is the absolute continuity of the Palm conditional probability ${P_0}$ relative to the absolute probability P on the $\sigma$-field ${\mathcal {F}_0}$. Further results are described in §1.