Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 47 / 1

A real weakly stationary process has finite second moments, a time-independent mean $\mu$, and autocovariance function ${\gamma }_k=\mathrm{Cov}(X_t,X_{t+k})$ depending only on the lag. Use the angular-frequency conventionThe time-series spectral density is nonnegative and integrates to ${\gamma }_0$. The displayed Fourier series is ordinarily justified by ${\sum }_k|{\gamma }_k|<\infty$. More generally, weak stationarity gives a spectral measure of a stationary time series, not necessarily a spectral density. If a spectral density exists without this summability, its Fourier inversion can be understood using Fejér sums, which converge in $L^1$ and at its Lebesgue points. Thus spectral density existence should not be inferred from weak stationarity alone.

A white noise process $WN(0,{\sigma }^2)$ has mean zero and $\mathrm{Cov}({\epsilon }_t,{\epsilon }_s)={\sigma }^2{1}_{\{t=s\}}$. Neither independence nor a normal distribution is needed for this second-order definition. Its autocovariance function is zero at nonzero lags, soFor a real absolutely summable filter $(c_r)$, the series defining $Y_t$ has mean-square convergence: its tails have $L^2$ norm at most $\Vert X_0{\Vert }_2{\sum }_{\text{tail}}|c_r|$. Its mean is $\mu {\sum }_r{c}_r$, andThe sum is absolutely convergent, since $|{\gamma }_X(j)|\le {\gamma }_X(0)$ by the Cauchy-Schwarz inequality. It depends only on $k$, proving that the output is a weakly stationary process. Substitute the spectral integral for ${\gamma }_X$ and interchange the sums and integral, justified by $f_X\in L^1$ and ${\sum }_{r,s}|c_r{c}_s|<\infty$. This givesBecause the coefficients are real, the two factors are complex conjugates. Hence the spectral density transformation under a linear filter isFor complex coefficients the same formula holds with the conjugate-covariance convention.

Applying this result to the order-two moving-average model yieldsEquivalently, its nonzero autocovariances are ${\gamma }_0={\sigma }^2(1+{\theta }_1^2+{\theta }_2^2)$, ${\gamma }_1={\sigma }^2{\theta }_1(1+{\theta }_2)$ and ${\gamma }_2={\sigma }^2{\theta }_2$, together with ${\gamma }_{-k}={\gamma }_k$.

For the order-one moving-average process, invert its filter by a geometric series. The required coefficients areThey are absolutely summable for $|\theta |<1$. Direct cancellation giveswith mean-square convergence. Thus the inverse filter recovers exactly the original white noise process, including its variance; it is an invertible time-series representation.