A characterization of E-processes and poisson processes in R n

Abstract

where Si are Borel sets and r i corresponding non-negative integers; (iii) P is a probability measure uniquely defined on 4. That this can actually be done is discussed in the book of Harris [6], p. 50ft. (o is a sequence of points of realization of a s.p.p. A s.p.p, is defined on an infinite B c R 2 in a similar manner. [N. B. N(S) will denote a r. v.; N(S, o~) will denote the actual value N(S) assumes for a given o).] Expectations taken w.r.t, the probability measure P will be denoted by the script ~. We note that (2 and ~ are the same for all s.p.p, and only the assignment of probabilities to events of the type (1.1.1) tells one s.p.p, from another. We shall therefore denote the above triple by P (in particular two different s.p.p, may be denoted by P~, P2 respectively). Two s.p.p, are equivalent, written P1 P2, if they have the same joint distribution for all events of type (1.1.1). Pr will be used as a generic symbol for "probabili ty". We shall also write c~= {xl}/=1, 2, . . . where the xl are arbitrarily labelled points in R 2 (without finite limit points). S will denote a bounded Borel set unless otherwise stated. A s.p.p. P is R 2 is stationary (in increments) if for every m-tuple of rectangles (a right-closed rectangle being a set of the form aj < xj < bj j = 1, 2) I /= 1, ..., m and every m-tuple r~ i=1, ..., m of non-negative integers we have, for every tER 2