A Note on Multivariate Poisson Flows on Stochastic Processes

In [1], a deterministic counting rate condition is shown to be necessary and sufficient for a counting process induced on a Markov step process Z to be multivariate Poisson. We show here that the result continues to hold without Z being a Markov step process. MARKOV STEP PROCESS It was assumed in [1] that a Markov step process Z induces a multivariate counting process N = tN«, N 2 , ••• , N c ) . The infinitesimal generator A of Z was used there to characterize a vector process whose respective components ri(Z(t)) can be heuristically interpreted as the counting rates for the corresponding N, at time t. It is shown in [1] that if the components of N do not have simultaneous jumps, a determinacy condition based on the sigma algebras Nt = u{N(u), u ~ t} is necessary and sufficient for N to consist of mutually independent Poisson processes. This condition is that for each t we have almost surely (1) E[r(Z(t)) INt] = E[r(Z(t))]. The above result is extended in the present letter to processes Z that need not be Markov. To this end, let Z be measurable with respect to an increasing family of sigma algebras {~}, and suppose further that Z induces the counting process N (as defined in [2], Chapter 2) in the sense that Nt ~Fr for each t. Let E[Ni(t)] nJ=r1[N;(u -) N,(s) = n,] dN,(u) for any 0 ~ s ~ t. (Equation (2), together with its possible implications, were called to the author's attention by Dr B. Melamed.) Moreover, Ni(t)-S~Ai(s)ds is not only an Ft-martingale, but also a fortiori an Nt-martingale. It then follows from the definition of Received 26 October 1982. * Postal address: Computer, Information and Control Engineering Program, The University of Michigan, Ann Arbor, MI 48109, U.S.A. 219 available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S000186780002111X Downloaded from https://www.cambridge.org/core. IP address: 54.191.40.80, on 20 Aug 2017 at 02:41:53, subject to the Cambridge Core terms of use, 220 Letters to the editor (4) intensity that on the right side of (2) (3) E[1' [N;(u -) N;(s) = n;] dN;(u) INs] = E[1' [Ni(u) Ni(s) = n;]Ai(U) du IN']' Equations (2) and (3) may be combined by taking the conditional expectation in (2) respective to N s, and substituting. If we then also add over ni = 0, 1,2, ... and apply Fubini's theorem, we obtain E[N;(t) N;(s) INs] =rE[Ai(U) INs] duo This equation effectively generalizes (1.18) of [1]; our Ai plays the role of the r, of [1],which in [1] is generated by a Markov step process Z. Indeed, under the assumptions of[1], our (4) specializes precisely to Equation (1.18) in [1].Condition (3.2) in [1] may be replaced by(5)E[Ai(t) INt] = E[Ai(t)]almost surely with respect to dt dP measure. As in [1], this condition (in the presence ofthe preceding hypotheses on N, N; Fr, and E[Ai(e) IN] above) is necessary and sufficientfor N to be a multivariate Poisson process respective to Nt. The proofs are easyexercises in the martingale theory of multivariate counting processes.If (5) is met, we have in (4)(6)E[Ai(U) IN;]= E{E[Ai(t) INt] INs} = E[Ai(t)].Thus N is a multivariate Poisson process according to the multichannel Watanabetheorem (see [2], Theorem II.T6). Conversely, let N be multivariate Poisson. From (4)and the Nt-independent increment property it follows that N, has the predictableNt-intensity E[Ai(·)]. But also, a version of E[Ai(·) IN] is such an intensity (see [2],Theorem II.T14). The uniqueness of predictable intensities ([2], Theorem II.T12) thenyields (5), as was desired. References[1] BEUTLER, F. J. AND MELAMED, B. (1982) Multivariate Poisson flows on Markov stepprocesses. J. Appl. Prob. 19, 289-300.[2] BREMAUD, P. (1981) Point Processes and Queues: Martingale Dynamics. Springer-Verlag,New York. available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S000186780002111XDownloaded from https://www.cambridge.org/core. IP address: 54.191.40.80, on 20 Aug 2017 at 02:41:53, subject to the Cambridge Core terms of use,