This exercise is related to the expression of the histogram (see Section 11.6).

(i) Show that (11.82) holds provided that F is twice continuously differentiable.

(ii) Show that the limit (11.81) is either zero or ∞ if F is replaced by ˆ F, the empirical d.f.

(iii) Show that the histogram is equal to the right side of (11.83) with probability 1.

(iv) Show that the histogram is pointwise consistent under the limiting process (11.84).