3.19 The purpose of this exercise is to construct a counterexample showing that it is possible that ????(V1|V2) < 0 even when V1 ⊂ V2. Let X(t) = √

2 cos(t +

Φ) denote a random cosine wave in one dimension, t ∈ ℝ1, where Φ is uniformly distributed on [0,2????). This process is stationary with covariance function ????(h) = cos(h). Set V = [0, a] for some a > 0.

(a) Show that var{X(V)} = 2(1 − cos a)∕a2

.

(b) In particular, note that X(V) = 0 and var{X(V)} = 0 when a is an integer multiple of 2????.

(c) Since var{X(V)} is not monotone decreasing in

a, it is possible for the dispersion variance to be negative. For example, show that

????([0,

a) | [0, b)) < 0 for a = 2????, b = 3????.

(d) Equation (3.65) states that the dispersion variance is always nonnegative when V2 = [0, b]is a union of n disjoint copies of V1 = [0, a], i.e. b =

na, a > 0, n ≥ 1. Using this result, deduce the trigonometric inequalities 1 − cos na ≤ n2

(1 − cos a)

for all real a > 0 and integer n ≥ 1.