Integrated random pulse process with positive and negative periodicity

The correlation parameter of renewal processes and structures with positive and negative periodicity

“…The spectral density of the strongly super-Poissonian RSs with a quasi-symmetric PDF of DBNP is independent of the shape of this PDF and have an identical Lorentzian line shape. The spectral width depends only on the periodicity parameter and is in agreement with (13). The strongly super-Poissonian structures are the most complicated for calculations, since there are different ways to obtain them.…”

“…If the correlation time is long, as it is for strongly sub-and super-Poissonian processes, the renewal process can theoretically be ergodic, but timeaveraging experimental data is impossible. The result also depends on the periodicity parameter, since sub-Poissonian processes quickly converge to the average even with a long correlation time [13].…”

“…This stationary stochastic point process can be represented as a collection of mathematical points randomly located on the time axis [4][5][6][7][8]. This process is also called the renewal pulse process [9][10][11][12][13]. The distances between neighboring points (DBNPs) are also called the waiting times, or holding times, or times between successive events.…”

“…The processes with equal correlation times may look different. Another criterion of randomness is the periodicity parameter introduced in [13,20]. This parameter turns to one for strongly sub-Poissonian processes, equals zero for the Poissonian process, and takes on negative values for super-Poissonian processes, which can therefore be assigned to a negative periodicity.…”

“…We consider PDFs of DBNP with interesting statistical properties. The shifted gamma-distribution is most widely used in physics and population dynamics [13,20,46]. The straightened trajectory of a molecule of an ideal gas with marked points of each nth collision is an example of such a structure.…”