Integrability of Expected Increments of Point Processes and a Related Random Change of Scale

Given a stationary point process with finite intensity on the real line R, denote by (Q Borel set in R) the random number of points that the process throws in Q and by <!-- MATH ${\mathcal{F}_t}(t \in R)$ --> the -field of events that happen in <!-- MATH $( - \infty ,t)$ --> . The main results are the following. If for each partition <!-- MATH $\Delta = \{ b = {\xi _0} < {\xi _1} < \cdots < {\xi _{n + 1}} = c\}$ --> <img width="317" height="41" align="MIDDLE" border="0" src="images/img6.gif" alt="$ \Delta = \{ b = {\xi _0} < {\xi _1} < \cdots < {\xi _{n + 1}} = c\} $"> of an interval [b, c] we set <!-- MATH ${S_\Delta }(\omega ) = \sum\nolimits_{\nu = 0}^n {E(N[{\xi _\nu },{\xi _{\nu + 1}})|{\mathcal{F}_{{\xi _\nu }}})}$ --> then <!-- MATH ${\lim _\Delta }{S_\Delta }(\omega ) = W(\omega ,[b,c))$ --> exists a.s. and in the mean when <!-- MATH ${\max _{0 \leqq \nu \leqq n}}({\xi _{\nu + 1}} - {\xi _\nu }) \to 0$ --> (the a.s. convergence requires a judicious choice of versions). If the random transformation <!-- MATH $t \Rightarrow W(\omega ,[0,1))$ --> of <!-- MATH $[0,\infty )$ --> 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 <!-- MATH ${\mathcal{F}_0}$ --> . The ratio <!-- MATH ${\varepsilon ^{ - 1}}E(N[0,\varepsilon )|{\mathcal{F}_0})$ --> converges a.s. as <!-- MATH $\varepsilon \downarrow 0$ --> . A necessary and sufficient condition for its convergence in the mean (as well as for the a.s. absolute continuity of the function on <!-- MATH $(0,\infty ))$ --> is the absolute continuity of the Palm conditional probability relative to the absolute probability P on the -field <!-- MATH ${\mathcal{F}_0}$ --> . Further results are described in §1.