非齐次Poisson过程的若干新性质

Some New Properties of Nonhomogeneous Poisson Process

本文在一些文献的基础上,进一步讨论非齐次Poisson过程的某些性质,给出了若干新性质。 设{N(t),t≥0}是累积强度为的非齐次Poisson过程。迄今的文献(如[1][2]等)指出,(?)n≥1,前n个到达时刻τ_1,τ_2,…,τ_n的联合概率密度为 本文定理1指出,上式不仅是必要的,而且是充分的,并给出了充分性的证明。从而,得到了描写过程统计规律的一个刻画。然后,在N(t_i)=n_i(i=1,…,k,0<t_1<…<t_k,0<n_1<…<n_k)的条件下,研究了过程的若干条件分布性质。定理2给出了位于各个时间间隔(t_(i-1),t_i](i=1,…,k)内的到达时刻τ_(n_i-1^(+1))…,τ_(n_i)的条件联合分布,证明了向量(τ_1,…,τ_n1)′,(τ_(n1+1),…,τ_(n2))′,(τ_(n_(k-1)^(+1)),…,τ_nh)′条件独立。定理3指出,过程{N(t),t_(j-1)<t<t_j}的有限维条件联合分布为多项分布(j=1,…,k),证明了k个过程{N(t),0<t<t_1},{N(t),t_1<t<t_2},…,{N(t),t_(k-1)<t<t_k)条件独立。

In this paper sonic properties of nonhomogeneous Poisson process and some new properties are discussed on the basis of some references.Let {N(t) t≥0} be the nonhomogeneous Poisson process with cumulativeintensity parameter . In references[1], [2], it was indicatedthat for Vn≥1, the joint distribution of the arrival times τ1, …, τn is In theorem 1 of this paper, it is indicated that the above expression is not only necessary but also sufficient, and the proof of sufficiency is given. Thus we obtain a characteristic condition which describes the statistical rule of the process. Then,under the conditions of N(ti)=ni (i=1, …, k, 0<t1<…<tk, 0<n1< … < nk), some properties of conditional distributions of the process are studied. In theorem 2, we obtain the conditional joint distribution of τni-1 + 1, …, τnj during the time interval (t1-1, ti] (i=1, …, k), and prove that the vectors (τ1, …, τn1)', (τn1 +1 , …., τn2)', …, (τnk-1+1, …, τnk)' are conditional independent. In theorem 3, we prove that the finite dimensional and conditional joint distribution of the process {N(t), ti-1<t<ti} is multinomial (j=1, …, k), and the processes{N(t), 0<t<t1}, {N(t), t1<t<t2}, …, (N(t), th-1<t<th} are conditional independent.