Are these statements of my professor about periodicity of harmonic processes in time series analysis correct? Assume $X_t$ is a harmonic stochastic process, i.e., $$X_t = \sum_{j=-k}^k A_j \exp(i \lambda_j t)$$ where the frequencies $\lambda_j$ are given and $A_j$ are uncorrelated random variables with zero mean and variance $\sigma_j^2$.
Then the spectral distribution $F$ of this process satisfies $$F(\lambda) = \sum_{\lambda \leq \lambda_j} \sigma_j^2,$$ i.e., it has finitely many jumps (at $\lambda_j$) and is constant elsewhere.

My professor now claims that the paths of this process are periodic (in a deterministic sense), i.e., $X_t=X_{t+T}$ a.s. for some $T$ and all $t$.

I think this is wrong. What about the harmonic process $$ X_t=A_1 \exp(i \frac{\pi}{2} t) + A_2 \exp(it) $$ which is harmonic in the sense above. The first part is periodic with period $4$, the last part with period $2\pi$. So the sum is not periodic, or is it? Where is my mistake? See also this post.
But it gets worse:

My professor claims that every process with a spectral distribution that has finitly many jumps and is constant elsewhere, has periodic paths (in a determinstic sense).

This can only be concluded from his previous arguments if any such process indeed is a harmonic process. But why is this the case? The spectral distribution does not in general determine the corresponding process uniquely, does it?
 A: Preliminary, let us do a slight change in notation: I will indicate with $k \in \mathbb{Z}$ the discrete time, with $K$ the $k$ present in the summation and the period (if it exists) with $N \in \mathbb{N}_0$.
Firstly, let us refresh the following concepts. Let $\theta \in \mathbb{R}_+$, $k \in \mathbb{Z}$, $i = \sqrt{-1}$, and consider the following (complex valued) function: $f(\theta) = e^{i \theta k}$. It is easy to check that this function is periodic with period $2\pi$, in fact
$
\begin{align}
f(\theta + 2\pi) = e^{i (\theta +2\pi) k} = e^{i \theta k} e^{i 2\pi k} = e^{i\theta k} = f(\theta)
\end{align}
$ 
since $e^{i 2 \pi k}=1$. Now, consider instead (with the same meaning of the symbols) the following (complex valued) function
$
\begin{align}
f(k) = e^{i \theta k}
\end{align}
$ 
Is it periodic? For discrete time functions, the period must be an integer, since the argument of the function must be an integer (the domain of $f(k)$ is $\mathbb{Z}$). A discrete time function can't be, e.g., periodic with period $\pi$. Thus, we must find, if it exists, an integer $N \in \mathbb{N}_0$ such as that $f(k+N) = f(k)$. Repeating the same reasoning as above,
$
\begin{align}
f(k+N) = e^{i\theta(k+N)} = e^{i \theta k} e^{i \theta N}
\end{align}
$ 
Now, $e^{i \theta N}$ is equal to $1$ if and only if $\ \theta N = 2 \pi n$, with $n \in \mathbb{N}$, which means:
$
\begin{align}
(*) \quad \theta = 2 \pi \frac{n}{N}, \quad n \in \mathbb{N}, N \in \mathbb{N}_0
\end{align}
$ 
The key concept is the following

The function $e^{i \theta k}$ is periodic (with respect to $k \in \mathbb{Z}$) if and
  only if the pulsation $\theta \in \mathbb{R}_+$ satisfies $(*)$, i.e., the pulsation is a rational multiple of $\pi$.


Now, back on your question. Your example is unfortunately ill-worded, since the period of discrete time functions/processes can't be $2 \pi$ ("the last part is periodic with period $2 \pi$" is, strictly speaking, wrong): as explained above, only integers are allowed. In fact, since $k$ is used as an (integer) index in the formalism $X_k$, what means, e.g., $X_{\pi}$ ? There exists only things like $X_1, X_2, X_3, \dots$, because your process is discrete time (this is the reason why I asked and, as it's now clear, it is a crucial fact).

Luckily enough, this means that it is very easy to prove that, in general, $X(k)$ (i.e., the process viewed as a function of discrete time) is not periodic, since the phasors are in general not periodic (with respect to $k$). However, I doubt your professor was referring to this: he probably was referring to something along the lines of $f(\theta)$ is $2 \pi$-periodic, i.e., viewing the process as a function not of time, but of the pulsations $\lambda_j$.

At this point, it is useless to describe in detail the fact about the spectral distribution. Thus, I will just give you a sketch. Using the Wiener-Khinchin theorem, we know that the spectral density of a random process is equal to the Fourier transform of the ACF (AutoCorrelation Function) of the process. But the ACF of a periodic function is periodic (and with the same period), thus the information on periodicity is "carried on" the spectral density.

This is the same as what happens deterministically. Let $f(t)$, $t \in \mathbb{R}$, be a function of (continuous) time; the Fourier transform of $f(t)$, which we can denote with $F(\omega)$, where $\omega \in \mathbb{R}$ is the pulsation, contains all the information present in $f(t)$. In short, this is simply the usual time domain - frequency domain duality.

Example: This is a trivial example to show that the process is in general not periodic. Starting with

$
\begin{align}
X_k = \sum_{j=-K}^{K} A_j e^{i \lambda_j k}
\end{align}
$ 
Now, choose $\lambda_j = \lambda \ \forall j$ (single frequency process), from which
$
\begin{align}
X_k = \sum_{j=-K}^{K} A_j e^{i \lambda k} = e^{i \lambda k} \underbrace{\sum_{j=-K}^{K} A_j}_{=A} = A e^{i \lambda k}
\end{align}
$ 
The periodicity of $X_k$ depends on the periodicity of the phasor $e^{i \lambda k}$, which depends on $\lambda$, remember the condition $(*)$. Thus, we can easily conclude that $X_k$ is, in general, not periodic.
A: It is possible that your professor is assuming all the frequencies in the sum are multiples of one, given, fundamental frequency and forgot to state this assertion.  If they are all multiples, then indeed almost all paths are periodic.
But even if they are not multiples, the paths are "quasi-periodic" in the sense of Harald Bohr (Niels Bohr's brother, a famous mathematician in his day): there exist infinitely many "quasi-periods" T in the sense that the particle returns to within epsilon of its value after T seconds.  There are subtle technicalities in the definitions here, I am being deliberately vague.  See the Collected Works of Norbert Wiener, vol. 2, the commentaries on his papers, pp. 102,111, 181, 325, 458.  Some engineers and physicists have been known to loosely use the word "periodic" when all they mean is "quasi-periodic", and they use the word "quasi-periodic" to mean something even more vague, I have seen an engineering textbook call the function $\sin x \over x$ "quasi-periodic".  Well!
