00 ), EN(O, t] = (EA)t, var N(O, t] = (EA)t + (var A)t 2 z EN(O, t], with strict inequality unless var A = 0. d. s with probability generating function g(s) = EsY (lsi ::5: 1), and let them be independent of a Poisson process No at rate A with N°(t) = N°(0, t]. Then =I N°(1) N(O, t] i=l Y; defines the counting function of a compound Poisson process for which EzN(O,tJ EN(O, t] var N(O, t] = exp[- At(l - g(z))], = A(EY)t, = A(var Y)t + A(EY) 2 t = [EW(t)](var Y) + [var W(t)](EY) 2 z EN(O, t], with strict inequality unless EY(Y- 1) = 0.

Rd. Rl, Pr{X > y} = e-uy; (ii) in IR 2 , Pr{X > y} = e-"lY 2 ; (iii) in IR 3 , Pr{X > y} = e-<4 "/3 llY'. These same expressions also hold for the nearest-neighbour distance of an arbitrarily chosen point of the process. 6. d. s with mean 1/Amax), yielding the points 0 < t 1 < t 2 <···,say. Then, independently for each k = 1, 2, ... , retain tk as a point of TI 1 with probability A(tk)/Amax and otherwise delete it. Verify that the residual set of points satisfies the independence axiom and that E( # {j: 0 < ti < u, tiE Tit}) = f: A(v)dv.

### An Introduction to the Theory of Point Processes by D.J. Daley, David Vere-Jones

by James

4.1