Background: We want to estimate the spectral density of a certain real signal $x(t)$, where time $t$ runs indefinitely over the real line (let's say that $x(t)$ may be the realization of a continuous stationary process). For q quick introduction, see this nice answer on PhysicsSE.
Following Wikipedia, the power $P$ of the signal is defined as:
$$ P= \lim_{T \rightarrow \infty } T^{-1}\int_{t_c-T/2}^{t_c+T/2} |x(t)|^2 dt = \lim_{T \rightarrow \infty } T^{-1}\int_{-\infty}^\infty |x_T(t)|^2 dt $$ where $t_c$ is some central time of the time window and $x_T$ is the continuation of $x$ by taking it to be zero outside the time window (case C in the figure): before taking the limit, $x(t)$ is effectively prolonged to be zero outside the time window of amplitude $T$. The corresponding power spectral density is $$ S(\omega) = \lim_{T \rightarrow \infty } T^{-1} |\hat{x}_T(\omega)|^2 $$
and, because of the Plancherel theorem,
$$ P = \int_{-\infty}^\infty S(\omega) d\omega $$
Question: Now, imagine we only have access to the function $x(t)$ over the time window $[t_0,t_0+T]$, as in the figure below. Therefore, it is not possible to take the limit $T\rightarrow \infty$ and we have to be content to calculate some "finite time" estimate of the "true" $P$ or the "true" $S$. What should we do? There is a list of methods on Wiki, but I see no clear and simple discussion of the effect of having a finite $T$.
In particular, we have various possibilities to continue the signal beyond the temporal window of size $T$. To me, the possibility B sounds quite "natural", in fact:
case A makes the signal periodic but introduces some jumps that are not there in the original signal (we assume $x(t)$ to be continuous).
case C is what (I imagine) people do in practice, namely we just consider $x_T(t)$. However, the signal is treated as an impulse of width $T$, while in reality, it is a continuous function $x(t)$ that is not necessarily impulsive.
Are different methods to estimate $S$ from $x_T(t)$ related somehow to the "choice" of how to extend $x_T(t)$ over the whole real line?
More precisely: Given the finite time window, we have no access to information on the part of the true spectral density $S(\omega)$ for $\omega \lesssim 1/T$: the observation should be longer than $T$ to probe these low frequencies. Therefore, it should be no surprise if A, B and C will disagree on this "infrared" part of the spectrum. I guess that the three estimates of $S(\omega)$ calculated by implementing A, B, C will be similar on some part of the $\omega$ domain. However, I expect the three prescriptions to give rise to different spectral leakages. Is there a way to estimate this? (or references where this estimate is outlined?).