On the event distance of Poisson processes with applications to sensors

We derive a closed formula for the expected distance between Poisson events of two i.i.d. Poisson processes with arrival rate λλ and respective arrival times X1,X2,…X1,X2,… and Y1,Y2,…Y1,Y2,… Namely, for any integers r≥0,k≥1r≥0,k≥1, the following identity holds: E[|Xk+r−Yk|]=k2−2k+1λ2kk(1+∑s=0r−1r−s(2k+s)2s⋅(2k+1)(s)(k+1)(s)), where x(q)x(q) denotes the Pochhammer polynomial. As a consequence we derive that the expected cost of a minimum weight matching with edges {Xi,Yi}{Xi,Yi} between two i.i.d. Poisson processes with arrival times X1,X2,…XnX1,X2,…Xn and Y1,Y2,…YnY1,Y2,…Yn is in Θ(n).