Relationship between time average and cross spectrum
I will infer from the "discrete Fourier coefficients" in your question that we are considering periodic signals, otherwise we should be considering a continuous spectrum (the Continuous-Time Fourier Transform). I will also be using the definition 
$$\left&lt; x(t), y(t) \right&gt;_t = \frac{1}{T}\int_{\alpha}^{\alpha +T} x^*(t) y(t) dt$$ where the superscript $*$ denotes complex conjugation, $T$ is the shortest period that both $x(t)$ and $y(t)$ share, and $\alpha$ is an arbitrary real number. If these definitions are not correct, please alter your question and provide the correct ones.

We know that we can decompose $x(t)$ and $y(t)$ using a Fourier series expansion
$$ x(t) = \sum_{n=-\infty}^{\infty} c_{x,n} \exp\left(\frac{2 \pi n t}{T}\right)$$
and
$$ y(t) = \sum_{n=-\infty}^{\infty} c_{y,n} \exp\left(\frac{2 \pi n t}{T}\right).$$

We know from Parseval's Theorem that 
$$ \sum_{n=-\infty}^{\infty} c^*_{x, n} c_{y, n} = \int_{-\frac{T}{2}}^{\frac{T}{2}} x^*(t) y(t) dt.$$ 
Notice the equality. Also notice that the right hand side is merely the product of $T$ and $\left&lt; x(t), y(t) \right&gt;_t$. This means that whenever the left hand side is zero, so is $\left&lt; x(t), y(t) \right&gt;_t$ and whenever $\left&lt; x(t), y(t) \right&gt;_t$ is zero so is the left hand side. I believe this resolves your question. Let me know if you have objections, and I will try to address them.