Show that XXXXXXXXXXimplies XXXXXXXXXXand that the converse may not hold. Verify this with the example where x2j−1 = −j, x2j = +j for j ≥ 1. Let Dn = X ⊤ n Xn and ∆n = diag(d 1/2 n11,... ,d1/2 npp)....

Show that (2.13) implies (2.14) and that the converse may not hold. Verify this with the example where x2j−1 = −j, x2j = +j for j ≥ 1.

Let Dn = X ⊤ n Xn and ∆n = diag(d 1/2 n11,... ,d1/2 npp). Assume that (2.14) holds and that the n −1dnjj , 1 ≤ j ≤ p are all bounded away from 0. Then show that as n → ∞,

                                  D1/2 n (βb n − β) D→ Np(0,σ2 Ip).