Fractal Time Poisson Process (FTPP)

• FTPP is a stochastic counting process characterized by independent inter-event times following the Mittag-Leffler law instead of the exponential distribution.
• It employs a time-change construction with inverse α-stable subordinators, leading to non-stationary increments and long-range dependence.
• The process is governed by fractional Kolmogorov equations with analytic solutions, facilitating Monte Carlo simulation and applications in anomalous diffusion, finance, and network traffic.

The Fractal Time Poisson Process (FTPP), also widely called the fractional Poisson process (FPP), is a stochastic counting process with independent and identically distributed inter-event times that follow the Mittag-Leffler distribution, as opposed to the exponential law of the classical Poisson process. The FTPP is neither a Markov process nor a Lévy process: its increments are non-stationary and non-independent, and its long memory manifests in anomalous statistical features suitable for modeling bursty phenomena characterized by heavy-tailed, power-law waiting times. The process is rigorously specified by an explicit renewal construction, fractional Kolmogorov equations, and analytic formulae for its finite-dimensional distributions, supporting both theoretical development and Monte Carlo simulation (Politi et al., 2011).

1. Time-Change Construction and Renewal Law

The canonical construction of the FTPP is by time-changing a standard rate-λ Poisson process, $N_1(\cdot)$, with the inverse of an independent strictly increasing α-stable subordinator $D_\alpha(u)$ ($0<\alpha\leq1$). Defining the inverse hitting-time process as $E_\alpha(t) = \inf\{u \geq 0: D_\alpha(u) > t\}$, the FTPP is given by:

$N_\alpha(t) := N_1(E_\alpha(t))$

Alternatively, $N_\alpha$ is a renewal process with i.i.d. interarrival times $\tau_j$ governed by the one-parameter Mittag-Leffler law:

$$\mathbb{P}(\tau \leq t) = 1 - E_\alpha(-\lambda t^\alpha)$$

where $$E_\alpha(z) = \sum_{n=0}^\infty z^n/\Gamma(\alpha n + 1)$$ is the Mittag-Leffler function (Politi et al., 2011).

2. Waiting-Time Distribution and Heavy-Tailed Behavior

The density of interarrival times in the FTPP is:

$$\psi_\alpha(t) = \lambda t^{\alpha-1} E_{\alpha,\alpha}(-\lambda t^\alpha)$$

with $D_\alpha(u)$0. Its Laplace transform is:

$D_\alpha(u)$1

For $D_\alpha(u)$2, the tail probability of $D_\alpha(u)$3 behaves as $D_\alpha(u)$4 for large $D_\alpha(u)$5, so the mean and all positive moments are infinite. This structure enforces power-law “bursty” waiting times, as opposed to the memoryless exponential distribution of the standard Poisson process. For $D_\alpha(u)$6, the Mittag-Leffler law reduces to the exponential law, reinstating the conventional Poisson process (Politi et al., 2011).

3. Governing Equations, State Distributions, and Generating Functions

Let $D_\alpha(u)$7. Then the process admits a governing fractional Kolmogorov system (Caputo sense):

$D_\alpha(u)$8

with initial condition $D_\alpha(u)$9. The Caputo derivative is:

$0<\alpha\leq1$0

Closed-form expressions for the state probabilities are given by:

$0<\alpha\leq1$1

and equivalently as:

$0<\alpha\leq1$2

The corresponding probability generating function is:

$0<\alpha\leq1$3

which solves the fractional PDE $0<\alpha\leq1$4 with $0<\alpha\leq1$5 (Politi et al., 2011).

4. Finite-Dimensional Distributions and Memory

The joint law for counts at a sequence of times $0<\alpha\leq1$6 is fully specified by a convolutional integral representation involving the waiting-time density and its convolutions: \begin{align*} \mathbb{P}(N_\alpha(t_1)=n_1, \dots, N_\alpha(t_m)=n_m) = \int_0^{t_1} du_1\, \psi_\alpha^{*n_1}(u_1) \int_{t_1-u_1}^\infty du_2\, \psi_\alpha(u_2) \cdots \ \times [1 - F_\alpha(t_m - \sum_{i=1}^{2m-1} u_i)] \end{align*} where $0<\alpha\leq1$7 denotes the $0<\alpha\leq1$8-fold convolution and $0<\alpha\leq1$9 is the cumulative distribution function of $E_\alpha(t) = \inf\{u \geq 0: D_\alpha(u) > t\}$0 (Politi et al., 2011).

Importantly, for $E_\alpha(t) = \inf\{u \geq 0: D_\alpha(u) > t\}$1, the process exhibits non-stationary, non-independent increments: the law of increments $E_\alpha(t) = \inf\{u \geq 0: D_\alpha(u) > t\}$2 depends on the age at $E_\alpha(t) = \inf\{u \geq 0: D_\alpha(u) > t\}$3, and the process is neither Markovian nor a Lévy process.

5. Simulation and Statistical Inference

For simulation, i.i.d. samples from the Mittag-Leffler density can be efficiently generated (e.g., Fulger et al. algorithm cited in (Politi et al., 2011)) by inverting a stable subordinator or by direct transformation:

• Sample $E_\alpha(t) = \inf\{u \geq 0: D_\alpha(u) > t\}$4
• Set

$E_\alpha(t) = \inf\{u \geq 0: D_\alpha(u) > t\}$5

and then use the cumulative sums for renewal epochs until surpassing a chosen time horizon.

Inference for $E_\alpha(t) = \inf\{u \geq 0: D_\alpha(u) > t\}$6 in empirical data may employ log–log tail plots (“stability plots”), QQ-plots against the theoretical Mittag-Leffler law, and likelihood-ratio tests for heavy-tailed departure from the exponential law (Hees et al., 2018).

6. Memory, Prediction, and Non-Markovian Effects

Unlike Poissonian waiting times, the Mittag-Leffler law is not memoryless. If $E_\alpha(t) = \inf\{u \geq 0: D_\alpha(u) > t\}$7 has already elapsed since the last event, the conditional density for the residual waiting time is:

$E_\alpha(t) = \inf\{u \geq 0: D_\alpha(u) > t\}$8

This form indicates that the longer the waiting time so far, the longer the expected further wait, with the hazard function strictly decreasing for $E_\alpha(t) = \inf\{u \geq 0: D_\alpha(u) > t\}$9. The process thus displays pronounced “aging” and long-term memory: bursts and silent periods are more likely to persist than in a Poisson process.

7. Applications, Impact, and Extensions

The FTPP is suited to modeling a range of real-world phenomena, including anomalous diffusion (e.g., as a scaling limit of continuous-time random walks with power-law trapping), blinking quantum dots in physics, financial time series with heavy-tailed transaction or waiting times, and bursty event series in geophysics and network traffic. It also serves as a paradigmatic model in the theory of anomalous statistical kinetics and non-Markovian renewal phenomena (Politi et al., 2011).

Extensions and generalizations include multivariate, non-homogeneous, compound, and higher-order fractional counting processes, as well as time-changed variants using general Lévy subordinators, furnishing further flexibility for modeling memory, overdispersion, and nonstationary features.

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References:

(Politi et al., 2011) – Full characterization of the fractional Poisson process (Hees et al., 2018) – Statistical inference for inter-arrival times of extreme events in bursty time series