I've been struggling with the following problem:
$\{X_t\}$ is a complex-valued stationary process such that $$ X_t = \sum_{j=1}^n A(\lambda_j) e^{it\lambda_j} $$ in which $-\pi < \lambda_1 < \lambda_2 < \cdots < \lambda_n = \pi$ and $A(\lambda_1),\cdots,A(\lambda_n)$ are uncorrelated complex-valued random coefficients (possibly zero) such that $$ \mathbb{E}[A(\lambda_j)] = 0 $$ and $$ \mathbb{E}[A(\lambda_j)\overline{A(\lambda_j)}] = \sigma_j^2 $$ Problem: Show that if ${X_t}$ is real-valued, it must hold that $A(\lambda_n)$ is real-valued, $\lambda_j = \lambda_{n-j}$ and $A(\lambda_j) = \overline{A(\lambda_{n-j})}$ for $j = 1, \cdots, n-1$.
I guess one way to prove it is to consider inner products of $X_t$ and $A(\lambda_j), j = 1, \cdots, n$, which cancel out many terms, but I just couldn't work it out.. The problem doesn't state clearly the index set, but I guess it's $t \in \mathbb{Z}$ according to the context. This seems to make the problem a bit more difficult :(
Any hint will be greatly appreciated!