3.19 The purpose of this exercise is to construct a counterexample showing that it is possible that...
Question:
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.
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