Question

(b) Yes. Consider the spectral density $S(f) = \pi \mathbb{1}\{f \in [-\frac{a}{2\pi}, \frac{a}{2\pi}]\}$. When $\tau = 0$, $s_0 = a = \int_{-1/2}^{1/2} S(f)df = \int_{-1/2}^{1/2} e^{2\pi if \cdot 0} S(f)df$. When $\tau \neq 0$, $s_\tau = \tau^{-1} \sin(a\tau) = \pi \int_{-\frac{a}{2\pi}}^{\frac{a}{2\pi}} \cos(2\pi f \tau)df = \pi \int_{-\frac{a}{2\pi}}^{\frac{a}{2\pi}} e^{2\pi if \tau} df = \int_{-1/2}^{1/2} e^{2\pi if \tau} S(f)df$. Hence \{ $s_\tau$ \} is an autocovariance sequence of a stationary time series.

          (b) Yes. Consider the spectral density $S(f) = \pi \mathbb{1}\{f \in [-\frac{a}{2\pi}, \frac{a}{2\pi}]\}$.
When $\tau = 0$,
$s_0 = a = \int_{-1/2}^{1/2} S(f)df = \int_{-1/2}^{1/2} e^{2\pi if \cdot 0} S(f)df$.
When $\tau \neq 0$,
$s_\tau = \tau^{-1} \sin(a\tau) = \pi \int_{-\frac{a}{2\pi}}^{\frac{a}{2\pi}} \cos(2\pi f \tau)df = \pi \int_{-\frac{a}{2\pi}}^{\frac{a}{2\pi}} e^{2\pi if \tau} df = \int_{-1/2}^{1/2} e^{2\pi if \tau} S(f)df$.
Hence \{ $s_\tau$ \} is an autocovariance sequence of a stationary time series.

$(b) Yes. Consider the spectral density S(f) = π1{f ∈ [-(a)/(2π), (a)/(2π)]}. When τ = 0, s0 = a = ∫-1/2^1/2 S(f)df = ∫-1/2^1/2 e^2π if · 0 S(f)df. When τ≠ 0, sτ = τ^-1sin(aτ) = π∫-(a)/(2π)^(a)/(2π)cos(2π f τ)df = π∫-(a)/(2π)^(a)/(2π) e^2π if τ df = ∫-1/2^1/2 e^2π if τ S(f)df. Hence { sτ } is an autocovariance sequence of a stationary time series.$

Added by Haley S.

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