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 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.