5. (40 points) Let N be a point process on Rk. For any Borel set AC Rk, let E[N(A)] = A(A). Assume that A(A) < if A is a bounded set. Define a rectangle to be a set of the form {x = (x1,...,xk) Rk : < ; < x; b,i = 1,...,k}. A Let me assume two properties. The first property is that for any collection of disjoint rectangles J,..., Jn, we have P{N(J) = 0,...,N(Jn) = 0} = e(A(J)+---+A(Jn)) The second property is that any rectangle can be split into two disjoint rectangles with the same expected number of points. Consider a rectangle J. To simplify the notation, let = A(J). For the first round of splitting, split J into 2 rectangles so that each interval has measure /2. For the second round of splitting, split each of those 2 rectangles so that we have 4 rectangles, each with measure /4. After k rounds of splitting, we have 2 rectangles with measure /2k. Let Mk be the number of non-empty rectangles among the 2k rectangles. By non-empty, I mean that an interval contains one or more points of the point process N. (a) Prove or disprove: Mk Mk+1 N(J) for k = N. (b) What is the distribution of Mk? Explain. (c) Compute limk E[Mk]. (d) What is the distribution of Moo limko Mk? Explain. (e) Can you show that N(J) = Moo? If so, what is the distribution of N(J)?