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2.1 Basic Properties 49
and
σ 2 θ ρ.
If |ρ| ≤ 1
2 , these equations give solutions θ (2ρ)−11 ± 1 − 4ρ2 and σ 2
2 1 + θ −1 1
. However, if |ρ| > , there is no real solution for θ and hence no MA(1)
2
process with ACVF κ. To show that there is no stationary process with ACVF κ,
we need to show that κ is not nonnegative definite. We shall do this directly from
the definition (2.1.5). First, if ρ> 1
2 , K [κ(i − j)]n i,j1 , and a is the n-component
vector a (1, −1, 1, −1,...) ′ , then
a ′ Ka n − 2(n − 1)ρ < 0 for n>2ρ/(2ρ − 1),
showing that κ(·) is not nonnegative definite and therefore, by Theorem 2.1.1, is not
, the same argument with a (1, 1, 1, 1,...)′
an autocovariance function. If ρ