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4.1 Spectral Densities 115
κ is an ACVF if and only if the function
f(λ) 1
∞
2π
h−∞
e −ihλ γ (h) 1
[1 + 2ρ cos λ]
2π
is nonnegative for all λ ∈ [−π, π]. But this occurs if and only if |ρ| ≤ 1
2 .
As illustrated in the previous example, Corollary 4.1.1 provides us with a powerful
tool for checking whether or not an absolutely summable function on the integers
is an autocovariance function. It is much simpler and much more informative than
direct verification of nonnegative definiteness as required in Theorem 2.1.1.
Not all autocovariance functions have a spectral density. For example, the stationary
time series
Xt A cos(ωt) + B sin(ωt), (4.1.6)
where A and B are uncorrelated random variables with mean 0 and variance 1, has
ACVF γ (h) cos(ωh) (Problem 2.2), which is not expressible as π
−π eihλf (λ)dλ,
with f a function on (−π, π]. Nevertheless, γ(·) can be written as the Fourier transform
of the discrete distribution function
⎧
⎪⎨
0 if λ