Strongly super-Poisson statistics replaced by a wide-pulse Poisson process: The billiard random generator

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

“…For µ = 0, this spectral density coincide with that in [20]. Figures 3 and 4 show the spectral densities for various parameters.…”

“…In the case of positive DBNP and long right tails, the spectral density is of the same shape (11) and the width ∆u super also depends on T /σ 2 , meanwhile the numerical coefficient is different. For the gamma distribution of DBNPs with small shape parameter, it is about 50 [20] instead of 4 (13).…”

“…The renewal process describes the registration of the interacting particles by a detector [14][15][16][17], collisions of billiard particles with the boundary [18][19][20], steps and moments of resetting in the search process [21][22][23][24][25], the nucleation events in a supersaturated nanophase [26], switching between two states of diffusivity [27], signals in a neuron network [28], the entry of particles into a living cell [29,30], animal population dynamics [31][32][33][34], networks of queues [35], and even special events in music [36]. The renewal theory has experienced a wave of new theoretical development, as well as some experimental progress in connection with the model of resetting a stochastic process (see [37] and references there).…”

“…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.…”