6.36 This exercise is associated with the proof of consistency of the LSE in Section 6.7, where Xni is defined below (6.82).

(i) Verify expression (6.82).

(ii) Show that (6.83) implies



n i=1 E(X2 ni ) → 0, which, in turn, implies (6.16) and (6.17); that is, in fact,

n i=1 E{Xni1(|Xni|≤τ)} −→0,

n i=1 var{Xni1(|Xni|≤τ)} −→0 for any τ > 0.

(iii) Show that (6.83) for every 1 ≤ j ≤ p is equivalent to (6.84).
(iv) Interpret the quantity 
n i=1(xi − ¯x)2 in Example 6.14.