Integrated random pulse process with positive and negative periodicity

A study of nonstationary processes that are integrals of stationary random sequences of delta pulses is presented. An integrated renewal process can be represented as the sum of a deterministic linear function of time and a Wiener process of the corresponding intensity. This intensity is determined by the mean value and variance of the waiting times of the pulse process and is greater for super-Poisson processes than for sub-Poisson ones. Linear growth over time of all cumulants is proved. An integrated random process with fixed time intervals can be replaced by the sum of a deterministic linear function and a random process with bounded variance. The analytical results are in good agreement with the numerical ones.

Figure 1

Sample contour for evaluating the Bromwich integral.

Figure 2

Plot of mean value and variance of pulse number vs time for RPP with super-Poisson statistics with different distributions of waiting times ($\langle \vartheta \rangle =1$, ${\sigma }_{\vartheta }^2=2$): shifted gamma distribution with $n=0.5$, $\alpha =2$, and ${\vartheta }_0=0$ (blue squares); shifted Weibull distribution with $n=0.5$, $\alpha =0.315$, and ${\vartheta }_0=0.37$ (red circles); Pareto distribution with $n=2.22$ and ${\vartheta }_0=0.55$ (green triangles). Analytical solution, obtained from (a) Eq. (24) (solid black line) and (b) Eq. (25) (solid black line), Eq. (A3) (dashed black line).

Figure 3

Plot of higher cumulants and measures of shape of pulse number vs time for RPP with super-Poisson statistics and shifted gamma distribution of waiting times with $n=0.5$, $\alpha =2$, and ${\vartheta }_0=0$. (a) Cumulants: ${\kappa }_3$ (black squares), ${\kappa }_4$ (red circles). Analytical solutions, obtained from Eq. (26) (solid black line) and Eq. (27) (solid red line). (b) Skewness ${\gamma }_1$ (black squares) and kurtosis excess ${\gamma }_2$ (red circles). Analytical solutions, obtained from Eq. (30) (solid black line) and Eq. (31) (solid red line).

Figure 4

Plot of mean value and variance of pulse number vs time for RPP with sub-Poisson statistics and different distributions of waiting times ($\langle \vartheta \rangle =1$, ${\sigma }_{\vartheta }^2=0.032$): shifted gamma distribution with $n=20$, $\alpha =0.04$, and ${\vartheta }_0=0.2$ (blue squares); shifted gamma distribution with $n=0.2$, $\alpha =0.4$, and ${\vartheta }_0=0.92$ (magenta down triangles); shifted Weibull distribution with $n=2$, $\alpha =0.384$, and ${\vartheta }_0=0.66$ (red circles); Pareto distribution with $n=6.67$ and ${\vartheta }_0=0.85$ (green up triangles). Analytical solution, obtained from (a) Eq. (24) and (b) Eq. (25) (solid black lines).

Figure 5

Probability density functions of waiting times for sub-Poisson process ($\langle \vartheta \rangle =1$, ${\sigma }_{\vartheta }^2=0.032$): shifted gamma distribution with $n=20$, $\alpha =0.04$, and ${\vartheta }_0=0.2$ (blue); shifted gamma distribution with $n=0.2$, $\alpha =0.4$, and ${\vartheta }_0=0.92$ (magenta); shifted Weibull distribution with $n=2$, $\alpha =0.384$, and ${\vartheta }_0=0.66$ (red); Pareto distribution with $n=6.67$ and ${\vartheta }_0=0.85$ (green).

Figure 6

Plot of mean value and variance of pulse number vs time for PPFTI with different distributions of deviations ($T=1$, ${\sigma }_{\vartheta }^2=1/6$): uniform distribution (48) with $\alpha =1$ (black squares); localized distribution (50) with $\alpha =1/\sqrt{3}$ (red circles). Analytical solutions, obtained from (a) Eq. (44) and (b) Eq. (45) (solid black and red lines, respectively).

Figure 7

Contour for evaluating the Bromwich integral for the Pareto PDF of waiting times. Crosses denote the simple poles of $\tilde{\mathcal{M}}(s,p)$; they shift to the left as $s\to 0$.