Let N, N1, N2 be simple point processes on a LCCB space (E,) such that N=N1+N2, and p() be a measurable function with 0<p()<1 on (E,). Then any two of the following statements yield another two:(Ⅰ) N is a Poiss...Let N, N1, N2 be simple point processes on a LCCB space (E,) such that N=N1+N2, and p() be a measurable function with 0<p()<1 on (E,). Then any two of the following statements yield another two:(Ⅰ) N is a Poisson process;(Ⅱ) N1 is the p( )thinning of N, N2 is the (1-p())-thinning of N;(Ⅲ) N1 and N2 are independent;(Ⅳ) N1, N2 are Poisson processes with respect to a filtration {F(A), A}, whereF(A)={N1(B), N2(B), B, BA},i.e., for each bounded set A, N1(A) and N2(A) are Poisson variables, independent of F(A ).Indeed, only the fact, (Ⅱ)+(Ⅲ)(Ⅳ)+(Ⅰ), is new.展开更多
文摘Let N, N1, N2 be simple point processes on a LCCB space (E,) such that N=N1+N2, and p() be a measurable function with 0<p()<1 on (E,). Then any two of the following statements yield another two:(Ⅰ) N is a Poisson process;(Ⅱ) N1 is the p( )thinning of N, N2 is the (1-p())-thinning of N;(Ⅲ) N1 and N2 are independent;(Ⅳ) N1, N2 are Poisson processes with respect to a filtration {F(A), A}, whereF(A)={N1(B), N2(B), B, BA},i.e., for each bounded set A, N1(A) and N2(A) are Poisson variables, independent of F(A ).Indeed, only the fact, (Ⅱ)+(Ⅲ)(Ⅳ)+(Ⅰ), is new.
