Let a rectangle partition consist of rectangles with side lengths hn1,...,hnd. Prove weak universal consistency for                        lim n→∞ hnj = 0(j = 1,...,d) and limn→∞ nhn1 ...hnd = ∞. A...

Let a rectangle partition consist of rectangles with side lengths hn1,...,hnd. Prove weak universal consistency for

                       lim n→∞ hnj = 0(j = 1,...,d) and limn→∞ nhn1 ...hnd = ∞.

A cell A is called empty if µn(A) = 0. Let Mn be the number of nonempty cells for Pn. Prove that under the condition (4.2), Mn/n → 0 a.s. Hint: For a sufficiently large sphere S consider 1 n n i=1 I{Xi∈S} for n → ∞.