Let us consider a MA($\infty$) process:
$y_t= \sum_{j=0}^{\infty} b_j \varepsilon_{t-j} $
where
$ \sum_{j=0}^{\infty} b_j^2 < \infty \>\>\>\>\>\> $ (1)
Given $f(\lambda)$ as the spectral density of the process $y_t$, does the square summabilty condition (1) imply that the following equality $\int_{-\pi}^\pi \log f(\lambda) d \lambda =0 $ holds ?
As a consequence of Theorem 5.81 in Brockwell and Davids (1986) "Time Series: theory and methods, second edition Springer", the relation $\int_{-\pi}^\pi \log f(\lambda) d \lambda = =0 $ holds for any stationary process $y_t$. In particular, if $y_t$ is long memory such that $ f(\lambda_c) = \infty$ for some $ \lambda_c \in [-\pi, \pi]$, then the previous relation is still true, because the theorem allows for a spectral density defined in $[-\pi, \pi]$ except for a set of zero Lebesgue measure.