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 ^ (t s R) the c-field of events that happen in ( -co, t).The main results are the following.If for each partition ,[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 ^o-The 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.