---
title: Fractal Time Poisson Process (FTPP)
url: https://www.emergentmind.com/topics/fractal-time-poisson-process-ftpp
type: topic
---

# Fractal Time Poisson Process (FTPP)

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 [1104.4234].

## 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 [1104.4234].

## 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 $E_{\alpha,\beta}(z) = \sum_{n=0}^\infty z^n/\Gamma(n\alpha + \beta)$. Its Laplace transform is:
$$
\mathcal{L}\{\psi_\alpha\}(s) = \frac{\lambda}{s^\alpha+\lambda}
$$
For $\alpha<1$, the tail probability of $\psi_\alpha$ behaves as $t^{-\alpha-1}$ for large $t$, 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 $\alpha=1$, the Mittag-Leffler law reduces to the exponential law, reinstating the conventional Poisson process [1104.4234].

## 3. Governing Equations, State Distributions, and Generating Functions

Let $p_k(t) = \mathbb{P}(N_\alpha(t) = k)$. Then the process admits a governing fractional Kolmogorov system (Caputo sense):
$$
D_t^\alpha\,p_k(t) = -\lambda p_k(t) + \lambda p_{k-1}(t), \quad k\geq 0
$$
with initial condition $p_k(0) = \delta_{k0}$. The Caputo derivative is:
$$
D_t^\alpha f(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t (t-s)^{-\alpha} f'(s)\, ds
$$
Closed-form expressions for the state probabilities are given by:
$$
p_k(t) = \frac{(\lambda t^\alpha)^k}{k!} \frac{d^k}{d(\lambda t^\alpha)^k} E_{\alpha,1}(-\lambda t^\alpha)
$$
and equivalently as:
$$
p_k(t) = \sum_{j=0}^\infty \frac{(-1)^j (\lambda t^\alpha)^{k+j}}{j! \Gamma((k+j)\alpha+1)}
$$
The corresponding probability generating function is:
$$
G(u,t) = \mathbb{E}[u^{N_\alpha(t)}] = E_{\alpha,1}(-\lambda (1-u)t^\alpha)
$$
which solves the fractional PDE $D_t^\alpha G = (u-1)\lambda G$ with $G(u,0)=1$ [1104.4234].

## 4. Finite-Dimensional Distributions and Memory

The joint law for counts at a sequence of times $0<t_1<\cdots<t_m$ 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 $\psi_\alpha^{*k}$ denotes the $k$-fold convolution and $F_\alpha$ is the cumulative distribution function of $\psi_\alpha$ [1104.4234].

Importantly, for $\alpha<1$, the process exhibits non-stationary, non-independent increments: the law of increments $N_\alpha(t+h)-N_\alpha(t)$ depends on the age at $t$, 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 [1104.4234]) by inverting a stable subordinator or by direct transformation:
- Sample $U, V \sim \text{Uniform}(0,1)$
- Set
$$
\tau = - (1/\lambda)^{1/\alpha} \ln U \left( \sin(\alpha\pi) \cot(\alpha\pi V) - \cos(\alpha\pi) \right)^{1/\alpha}
$$
and then use the cumulative sums for renewal epochs until surpassing a chosen time horizon.

Inference for $\alpha, \lambda$ 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 [1802.05218].

## 6. Memory, Prediction, and Non-Markovian Effects

Unlike Poissonian waiting times, the Mittag-Leffler law is not memoryless. If $t_0$ has already elapsed since the last event, the conditional density for the residual waiting time is:
$$
p(t|t_0) = \frac{\psi_\alpha(t+t_0)}{1 - F_\alpha(t_0)}
$$
This form indicates that the longer the waiting time so far, the longer the expected further wait, with the hazard function strictly decreasing for $\alpha<1$. 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 [1104.4234].

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.

---

**References:**  
[1104.4234] – Full characterization of the fractional Poisson process  
[1802.05218] – Statistical inference for inter-arrival times of extreme events in bursty time series

Source: https://www.emergentmind.com/topics/fractal-time-poisson-process-ftpp