Asymptotic Analysis of Discrete-Time Hawkes Process

This paper investigates the discrete-time Hawkes process by establishing its Large Deviation Principle, analyzing the limiting behavior of the associated arrival process, and demonstrating its application in modeling insurance claims.

The Gist

Imagine you are running a busy coffee shop. Usually, customers arrive randomly, like raindrops hitting a roof. But in this paper, the authors are studying a very specific, chaotic kind of coffee shop: The Self-Exciting Coffee Shop.

In this shop, every time a customer walks in, they don't just order coffee; they also shout, "Hey, look who's here!" This noise makes other people more likely to walk in immediately after. One arrival triggers another, which triggers a third, creating a sudden, massive rush of customers. In the real world, this happens with earthquakes (one quake triggers aftershocks), stock market crashes (one sell-off triggers more), or viral social media posts.

The authors of this paper are mathematicians who wanted to understand the long-term behavior of these "rushes" when time is measured in steps (like minutes or days) rather than a smooth, continuous flow.

Here is the breakdown of their work using simple analogies:

1. The Setup: The "Trigger" and the "Count"

The paper looks at two things:

• The Arrival Process ($\xi$): A light switch that is either ON (1) or OFF (0). If it's ON, a customer arrives. If OFF, no one comes.
• The Hawkes Process ($H$): A counter that keeps a running total of how many customers have arrived so far.

The magic rule is: The more people who have arrived in the past, the higher the chance that someone will arrive right now. It's like a snowball rolling down a hill; the bigger it gets, the more snow it picks up, and the faster it grows.

2. The Big Question: What happens in the long run?

The authors asked: "If we watch this coffee shop for a very, very long time, what will the average number of customers look like?"

• The "Law of Large Numbers" (The Average): They proved that even though the arrivals are chaotic and trigger each other, the average number of customers per minute eventually settles down to a predictable number. It's like saying, "Even though the rush hour is crazy, on average, 5 people walk in every minute."
• The "Central Limit Theorem" (The Fluctuations): They also showed that if you look at how much the actual number of customers deviates from that average, it follows a familiar "bell curve" pattern.

3. The Big Discovery: The "Rare Event" Rule (Large Deviation Principle)

This is the most important part of the paper. Usually, we care about the average behavior. But what if something weird happens? What if the coffee shop suddenly gets 100 times more customers than usual? Or what if it goes completely empty for an hour?

In probability, these are called "Rare Events." The authors developed a mathematical tool called the Large Deviation Principle (LDP).

• The Analogy: Imagine you are betting on a coin flip. Getting 5 heads in a row is rare. Getting 100 heads in a row is extremely rare. The LDP is a formula that tells you exactly how unlikely a specific crazy event is.
• Why it matters: It gives a "speed limit" for how fast the probability of a disaster drops. It tells you: "The chance of this massive crowd happening is so small, it's roughly equal to $e^{-1000}$." This is crucial for risk management.

4. The Real-World Application: The Insurance Company

To show why this math is useful, the authors applied it to an Insurance Company.

• The Scenario: An insurance company collects a small fee (premium) from everyone every day. But sometimes, people file claims (the "arrivals").
• The Danger: If claims happen too often (a "self-exciting" rush of claims, like after a natural disaster), the company might run out of money and go bankrupt.
• The Solution: Using their new math, the company can calculate the exact minimum premium they need to charge to stay safe.
  • If they charge too little, the "rush" of claims will eventually drain their funds.
  • If they charge just enough (based on the formula in the paper), they can survive even the rare, massive rushes of claims.
• The "Ruin Probability": The math also lets them calculate the tiny, tiny chance that they will still go bankrupt even if they charge the right amount. It's like knowing the odds of your house burning down even if you have a sprinkler system.

5. The "Feasible Region" (The Safety Zone)

The authors drew some graphs (Figure 1 and 3) that look like a sandwich.

• The top slice of bread is the "worst-case scenario" (maximum possible chaos).
• The bottom slice is the "best-case scenario" (minimum chaos).
• The filling (the yellow/blue shaded area) is where the reality must live.
  This helps businesses know the boundaries of risk. They know the chaos can't get worse than the top line or better than the bottom line.

Summary

In plain English, this paper says:

"We have figured out the rules for how 'contagious' events (like crowds, earthquakes, or stock crashes) behave over time. We can now predict the average behavior, but more importantly, we can calculate the exact odds of a disaster. This allows insurance companies and banks to set their prices and safety nets perfectly, ensuring they don't go broke when the unexpected happens."

It turns the chaos of "one thing triggering another" into a predictable, manageable risk.

Technical Summary

Here is a detailed technical summary of the paper "Asymptotic Analysis of Discrete-Time Hawkes Process" by Utpal Jyoti Deba Sarma and Dharmaraja Selvamuthu.

1. Problem Statement and Motivation

The paper addresses the gap in the theoretical understanding of Discrete-Time Hawkes Processes (DTHP). While continuous-time Hawkes processes are well-studied in fields like finance and seismology due to their self-exciting and clustering properties, many real-world datasets are recorded in discrete time or aggregated forms. In such contexts, continuous models may be inefficient or inapplicable.

Although previous studies (Sarma & Selvamuthu, Seol) had established the Law of Large Numbers (WLLN, SLLN) and Central Limit Theorem (CLT) for DTHP, the Large Deviation Principle (LDP)—which characterizes the probability of rare events and tail behavior—had not been rigorously established for the standard discrete-time model defined by Seol. This paper aims to:

1. Analyze the asymptotic behavior of the underlying arrival process $\{\xi_n\}$.
2. Establish the LDP for the DTHP $\{H_n\}$.
3. Derive bounds for the scaled logarithmic moment generating function (MGF).
4. Demonstrate a practical application in insurance risk modeling.

2. Methodology and Model Definition

The Model

The authors define a discrete-time Hawkes process based on a sequence of Bernoulli random variables $\{\xi_n\}_{n \geq 1}$ taking values in $\{0, 1\}$.

• Exciting Function: A sequence of positive real numbers $(a_i)_{i=0}^\infty$ satisfying $\sum a_i < 1$ and $\sum i a_i < \infty$.
• Intensity: The probability of an arrival at time $n$ depends on the entire past history:
  $$P(\xi_n = 1 | \xi_1, \dots, \xi_{n-1}) = a_0 + \sum_{i=1}^{n-1} a_{n-i}\xi_i$$
• DTHP: The cumulative count of events up to time $n$ is $H_n = \sum_{i=1}^n \xi_i$.

Mathematical Tools

The analysis relies heavily on Large Deviation Theory and specific functional analysis techniques:

• Gärtner-Ellis Theorem: Used to establish LDP via the limit of the scaled logarithmic MGF, though the authors note the difficulty in proving differentiability.
• Bryc's Theorem: The primary tool used to prove LDP. It requires:
  1. Exponential Tightness of the distributions of $H_n/n$.
  2. Existence of the limit of the scaled logarithmic MGF for a well-separating collection of functions.
• Nearly Subadditive Sequences: Theorem 2.4 (de Bruijn and Erdős) is utilized to prove the convergence of the scaled logarithmic MGF by bounding the error terms in subadditivity inequalities.
• Associated Random Variables: The property that the $\xi_i$ are associated (positive covariance for monotone functions) is used to derive bounds for the MGF.

3. Key Contributions and Results

A. Asymptotic Behavior of the Arrival Process

The authors prove that the arrival process $\xi_n$ converges in distribution to a Bernoulli random variable $\xi$ as $n \to \infty$.

• Result: $\xi_n \xrightarrow{d} \xi$, where $P(\xi=1) = \frac{a_0}{1 - \sum_{i=1}^\infty a_i}$.
• Significance: This establishes the long-term stationary probability of an event occurring, which is crucial for defining the rate function in the LDP.

B. Establishment of the Large Deviation Principle (LDP)

The paper proves that the sequence of normalized DTHP $\{H_n/n\}$ satisfies the LDP with a good rate function $R(x)$.

• Method: They verify the conditions of Bryc's Theorem:
  1. Exponential Tightness: Proven by showing $H_n/n$ is bounded within $[0, 1]$.
  2. Limit Existence: They prove the existence of the limit $\Gamma_g = \lim_{n \to \infty} \frac{1}{n} \log E[e^{n g(H_n/n)}]$ for a well-separating collection of concave continuous functions.
• Rate Function: The rate function is identified as the Fenchel-Legendre transform of the limit function $\Gamma(t)$: $R(x) = \sup_{t} \{tx - \Gamma(t)\}$.

C. Convergence and Bounds of the Scaled Logarithmic MGF

The authors prove the pointwise convergence of the scaled logarithmic MGF:
$$\Gamma(t) = \lim_{n \to \infty} \frac{1}{n} \log E[e^{t H_n}]$$
They establish that $\Gamma(t)$ is convex and differentiable almost everywhere. Crucially, they derive explicit upper and lower bounds for $\Gamma(t)$:

• Lower Bound ($L(t)$): Derived using the limit of the arrival probability and the initial probability of no events.
• Upper Bound ($U(t)$): Derived using the sum of the exciting function coefficients.
• Significance: These bounds allow for the estimation of the rate function $R(x)$ without needing an explicit closed-form solution, which is often intractable for complex self-exciting processes.

D. Application: Insurance Surplus Process

The paper applies the theoretical results to a discrete-time risk model for an insurance company.

• Model: Surplus $U_n = u + np - H_n$, where $u$ is initial capital, $p$ is premium, and $H_n$ is the number of claims.
• Profitability Condition: Using the SLLN, they derive that the company remains profitable in the long run if $p > \frac{a_0}{1 - \sum a_i}$.
• Ruin Probability: Using the LDP, they approximate the probability of ruin (surplus becoming negative) for large $n$:
  $$P(U_n < 0) \approx e^{-n \inf_{x \in (u/n + p, 1)} R(x)}$$
  This quantifies the risk of bankruptcy even when the premium is set above the break-even point.

4. Significance and Impact

1. Theoretical Advancement: This work fills a critical gap in the literature by extending Large Deviation Theory to the discrete-time Hawkes process, a model increasingly relevant for digital and aggregated data.
2. Methodological Rigor: The use of Bryc's Theorem combined with the concept of nearly subadditive sequences provides a robust framework for handling the dependencies inherent in self-exciting processes where standard independence assumptions fail.
3. Practical Utility: The application to insurance demonstrates how the LDP can be used to quantify rare event risks (bankruptcy) in a self-exciting environment. Unlike the CLT, which describes typical fluctuations, the LDP provides the exponential decay rate of tail probabilities, which is essential for risk management and capital reserve calculations.
4. Numerical Validation: The paper includes Monte Carlo simulations that verify the theoretical profitability conditions and illustrate the behavior of the surplus process under different premium settings, confirming the theoretical bounds.

Conclusion

The paper successfully characterizes the asymptotic behavior of the discrete-time Hawkes process. By proving the LDP and deriving bounds for the limit function, the authors provide a powerful tool for analyzing rare events in self-exciting systems. The application to insurance risk modeling highlights the practical value of these theoretical results, offering a method to calculate ruin probabilities in a setting where claim arrivals are clustered and history-dependent. Future work suggested by the authors includes finding an explicit expression for the rate function and incorporating mutual excitation/inhibition mechanisms.