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Long-memory Markov chains with power-law intensities

This paper introduces a self-exciting point process with power-law intensity dynamics that utilizes a finite-dimensional nonlinear Markov chain to approximate long-memory behavior while maintaining global stability and a unique invariant distribution under specific conditions.

Original authors: Kyungsub Lee

Published 2026-07-24

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Kyungsub Lee

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Rhythm of Randomness

Imagine you are watching a busy city street. Cars, pedestrians, and delivery trucks zip by in a chaotic dance. Sometimes, a single event triggers a cascade: a red light turns, and suddenly a whole line of cars stops; a street performer starts juggling, and a crowd gathers, slowing down traffic even more. In science, this is called a "self-exciting" process, where one event makes the next one more likely to happen soon. For decades, mathematicians have used a tool called the Hawkes process to model this. Think of it as a digital drumbeat: every time a drum is hit, the sound gets louder for a moment before fading away.

The tricky part is how that sound fades. In many simple models, the sound fades away quickly, like a bell that stops ringing after a few seconds. This is easy to predict and calculate. But in the real world—like in stock markets, earthquakes, or social media trends—the "echo" often lingers much longer, fading slowly like a giant bell that keeps humming for hours. This is called "long memory." The problem is that when the echo fades slowly (following a "power law"), the math gets incredibly messy. To predict the next event, you theoretically need to remember every single event that has ever happened, stretching back to the beginning of time. It's like trying to drive a car while looking at a rearview mirror that shows the entire history of the universe; it's too much information to handle. This paper steps into that messy corner of statistics to see if we can build a simpler, smarter way to track these long echoes without needing a supercomputer to remember everything.

The Paper's Big Idea: A Memory-Saving Trick

The authors, led by Kyungsub Lee, have built a new kind of mathematical model that acts like a "long-memory" system but is actually much easier to use. They call it a Markovian intensity model. To understand why this is a big deal, imagine you are trying to predict when the next person will walk into a coffee shop.

In the old, complicated way (the standard power-law Hawkes process), to know the probability of the next customer arriving, you would need to know exactly when every single previous customer walked in, because each one adds a tiny bit of "excitement" that fades away slowly. It's like trying to calculate the temperature of a room by remembering every time the heater was turned on for the last 100 years.

Lee's new model is a clever shortcut. Instead of remembering the entire history of the coffee shop, the model only keeps track of two numbers (a "2-dimensional state") at any given moment:

How loud the "noise" is right now (the current intensity).
How fast that noise is fading (the slope of the decay).

Think of it like a video game character who has a "health bar" and a "regeneration rate." You don't need to know every time the character took damage in the past; you just need to know their current health and how fast they are healing. The authors proved that by updating these two numbers every time a new event happens, the model can perfectly mimic the "long memory" behavior of the old, complicated system. It captures the same slow-fading echoes but does so with a tiny, manageable memory.

What They Found and Proved

The paper doesn't just guess that this trick works; they put it through a rigorous mathematical workout.

First, they showed that this new system is stable. In the world of math, "stable" means the system doesn't go crazy. If you run this model for a long time, the "health bar" and "regeneration rate" won't explode to infinity or crash to zero; they will settle into a predictable, repeating pattern. The authors proved this using a set of rules called "Lyapunov criteria," which are like checking if a ball rolling down a hill will eventually stop in a valley rather than flying off a cliff. They found that as long as the "excitement" from new events isn't too strong compared to the natural fading speed, the system stays calm and predictable.

Second, they showed that the model is truly random but connected. They proved that the system can reach any possible state (it's "irreducible") and doesn't get stuck in a boring loop (it's "aperiodic"). This means the model is flexible enough to handle the messy, unpredictable nature of real-world events while still being mathematically sound.

Finally, they ran simulations to see if the model actually behaves like the real world. They generated 50,000 fake events using their new model and analyzed them with a tool called the "Local Whittle estimator." This tool is like a detective that looks for hidden patterns in the noise. The results showed that when the model is tuned to the edge of its stability (where things are most intense), it successfully reproduces the "long-memory" behavior found in nature. The "echo" of past events lingered just as long as it should, confirming that the two-number shortcut works.

Why This Matters

The beauty of this paper is that it solves a practical headache. Before this, if you wanted to model a system with long-lasting echoes (like financial crashes or viral trends), you had to choose between accuracy (using the messy, history-heavy model) or speed (using a simple model that forgets too much). Lee's model gives you the best of both worlds: the accuracy of the long-memory system with the speed and simplicity of a system that only remembers two numbers.

The authors are careful to note that these results are based on simulations and mathematical proofs, not real-world data collection yet. However, they have shown that the theory holds up. By replacing the need to remember the entire history of the universe with a simple, two-number update rule, they have opened the door for easier, faster, and more accurate predictions of complex, self-exciting events in our world. It's a reminder that sometimes, to understand the long past, you don't need to look back at everything; you just need to know the right two things about right now.

Technical Summary: Long-memory Markov chains with power-law intensities

Problem Statement
The paper addresses a fundamental limitation in the modeling of self-exciting point processes, specifically Hawkes processes with power-law kernels. While power-law Hawkes processes effectively capture long-memory behavior and slow temporal decay observed in natural and social systems, they lack a finite-dimensional Markovian representation. Unlike exponential-kernel Hawkes processes, which admit a Markov formulation via a simple intensity state, power-law kernels induce dependence on the entire event history. This non-Markovian structure complicates theoretical analysis, likelihood-based inference, simulation, and control, often forcing researchers to rely on approximations such as kernel truncation or sums of exponentials.

Methodology
The authors propose a novel self-exciting point process that combines long-memory behavior with a finite-dimensional Markovian state representation. The core methodology involves the following steps:

Model Construction: The model retains the local jump and slope update structure of power-law Hawkes processes at event times. Specifically, at each event time $T_n$ , the conditional intensity $\lambda(t)$ experiences a constant upward jump and its slope experiences a constant downward jump, consistent with power-law excitation.
Latent State Augmentation: To eliminate global history dependence, the authors augment the state space to include a two-dimensional latent vector $(X_n, c_n)$ $(X_{n}, c_{n})$ .
- $X_n$ represents a scaled intensity component.
- $c_n$ acts as a scale parameter governing the decay dynamics.
- Between events, the conditional intensity decays according to a power law driven by these latent states: $\lambda_n(t) = \mu + X_n (1 + t/c_n)^{-p}$ .
State Updates: The latent states are updated at each arrival based on the inter-arrival time $\tau_n$ . The update rules are derived to ensure the local intensity and slope match the power-law Hawkes dynamics, resulting in a nonlinear stochastic system driven by exogenous inter-arrival times.
Theoretical Framework: The process is analyzed within a general state-space Markov chain framework (specifically a Nonlinear State-Space model, NSS). The authors employ the general theory of Markov chains developed by Meyn and Tweedie (2009) to establish:
- Irreducibility and Aperiodicity: By analyzing the associated deterministic control model (CM), they prove the existence of a global attracting state $(\xi, \gamma)$ and demonstrate that the system is forward accessible and M-irreducible.
- Stability: Using a Foster-Lyapunov criterion, they derive explicit stability conditions on the model parameters ( $\mu, \gamma, \xi, p$ ) to ensure the latent state process is positive Harris recurrent. This guarantees the existence of a unique invariant distribution.
Estimation and Simulation: The paper utilizes the Local Whittle Estimator (LWE) on simulated data to estimate the memory parameter $d$ of the log inter-arrival times, assessing the model's ability to reproduce long-memory behavior.

Key Contributions

Finite-Dimensional Markovian Representation: The primary contribution is the construction of a 2-dimensional Markov chain that reproduces the key features of power-law self-excitation without requiring the full event history.
Rigorous Stability Analysis: The paper provides explicit stability conditions (Assumption 1) under which the latent process is positive Harris recurrent, ensuring long-run stability and the existence of a unique invariant measure.
Theoretical Characterization: The authors establish that the proposed process is a $\psi$ -irreducible, aperiodic T-chain, providing a solid theoretical foundation for inference and simulation.
Empirical Validation: Simulation results demonstrate that the model exhibits long-memory behavior, particularly near the boundary of the stability region, as quantified by the Local Whittle Estimator.

Results

Theoretical: The paper proves that under mild regularity conditions on the inter-arrival time distribution and specific parameter constraints (involving $\xi, p, \gamma$ ), the embedded Markov chain is positive Harris recurrent. This ensures the process does not explode and converges to a stationary distribution.
Simulation: Using a sample size of $n=50,000$ $n = 50, 000$ with unit exponential inter-arrival noise, the authors varied parameters $p$ $p$ and $\xi$ $ξ$ to explore different regimes.
- The estimated memory parameter $\hat{d}$ increases with the excitation parameter $\xi$ , indicating stronger persistence.
- Estimates near the stability boundary approach the theoretical upper limit of $0.5$ (the threshold for stationarity), consistent with near-critical dynamics.
- The Local Whittle Estimator showed robustness across different bandwidth choices, confirming that the observed power-law behavior is a structural feature of the model rather than an artifact of estimation.

Significance and Claims
The paper claims to offer a tractable continuous-time formulation for modeling long-memory self-exciting processes. By replacing explicit history dependence with a fixed-dimensional latent Markov state, the model enables exact simulation and likelihood-based inference, which are often intractable for standard power-law Hawkes processes. The authors position this work as a bridge between the empirical appeal of power-law decay and the analytical convenience of Markovian dynamics. The significance lies in providing a mathematically rigorous, stable, and computationally feasible alternative for modeling event-time dynamics in systems exhibiting long-range dependence, without resorting to the approximations or infinite-dimensional representations typically required for power-law kernels.

The Rhythm of Randomness

The Paper's Big Idea: A Memory-Saving Trick

What They Found and Proved

Why This Matters

Technical Summary: Long-memory Markov chains with power-law intensities

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