Comparing FFT for increasing time intervals in a signal
I'm afraid the premise of your approach is faulty as can be shown with a single counterexample.  Suppose your signal is a simple real pure tone sinusoidal.  When the interval length reaches 3 cycles, bin index 3 (zero based) of the DFT will be non-zero and all the rest of the DFT bins will be zero.  (Considering only the lower half of the DFT values because a real valued signal as the upper half is a conjugate mirror of the lower half.)  Then as your interval increases the DFT starts to smear with leakage.  When you reach 3 and 1/2 cycles, the smearing will be the greatest and the magnitudes of bins 3 and 4 will be nearly equal and their phases nearly 180 degrees apart.  When you get up to 4 cycles in your sampling interval, once again all the bin values are zero except for bin 4.  That behavior will repeat for each whole number of cycles your interval reaches and therefore there will never be any kind of convergence happening.

It is not clear to me from your question, but are you assuming your signals are mixes of steady pure tones?