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Irregularly observed long-memory Levy-driven moving average processes

This paper investigates the asymptotic behavior of normalized partial sums for long-memory continuous-time moving-average processes driven by a Lévy process when observed at random, irregular renewal times under both finite- and infinite-mean sampling schemes.

Original authors: Mohamedou Ould Haye, Anne Philippe

Published 2026-08-04

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Mohamedou Ould Haye, Anne Philippe

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

Imagine you are trying to understand the rhythm of a chaotic drumbeat. In the world of statistics, this drumbeat is a "time series"—a sequence of data points recorded over time, like stock prices, weather patterns, or heartbeats. Usually, scientists assume these beats happen at regular intervals, like a metronome ticking every second. But in the real world, things are rarely so neat. Sometimes you check your phone for a signal only when you walk into a room; sometimes a telescope captures a star only when the clouds part. These are "irregularly observed" events, where the time between measurements is random and unpredictable.

To make sense of these messy, irregular data points, mathematicians use a powerful tool called a "moving average process." Think of this as a smooth, flowing river that carries a memory of everything that happened upstream. If the river has "long memory," a drop of water from a long time ago still influences the current flow today. This paper focuses on a specific type of river driven by "Lévy processes," which are like the river's source, capable of sudden, wild jumps (like a storm) rather than just a gentle, steady rain. The big question the authors tackle is: if we dip our buckets into this river at completely random, irregular times, does the water we collect still behave like a predictable, smooth river, or does the randomness of our bucket-dipping turn the whole thing into a chaotic mess?

This paper, written by Mohamedou Ould Haye and Anne Philippe, dives deep into this puzzle. They study what happens when you take a long-memory river driven by these wild Lévy jumps and sample it at random moments defined by a "renewal process"—essentially, a sequence of random waiting times between observations. The authors wanted to know: if we add up the water we collected (the "partial sums"), what kind of shape does that total take as we collect more and more buckets?

The researchers found that the answer depends entirely on how "heavy" the tails of the waiting times are. If the waiting times between observations have a finite average (meaning you don't wait an eternity between checks), the total amount of water collected eventually settles into a familiar, bell-shaped curve known as a normal distribution. It's as if the randomness of the sampling smooths itself out, and the result looks like a standard, predictable bell.

However, the story gets much more interesting if the waiting times are "heavy-tailed," meaning there is a significant chance of waiting an extremely long time. In this scenario, the math reveals a surprising twist. If the waiting times are heavy enough, the total amount of water doesn't just form a simple bell curve. Instead, it forms a "normal variance mixture." Imagine a bell curve, but the width of that bell is controlled by a random, wobbly variable. This random width is determined by a complex dance between how "heavy" the waiting times are and how "long" the river's memory stretches. The authors proved that in these specific heavy-tail conditions, the final result is a standard bell curve multiplied by a random factor derived from a "stable subordinator"—a mathematical object that acts like a random, jagged clock ticking in the background.

The paper rigorously proves these outcomes using a mix of advanced calculus and probability theory. They established a new "Lindeberg proposition" for continuous-time processes, which acts like a rulebook for when a chaotic sum of random events will calm down into a normal distribution. They showed that while the final results look similar to what was known for discrete, step-by-step data, the path to proving it for continuous, flowing time is far from routine. The authors didn't just guess; they provided a complete mathematical proof that covers both the "finite-mean" (regular) and "infinite-mean" (heavy-tailed) cases, confirming exactly how the irregular sampling reshapes the memory of the underlying process.

Technical Summary: Irregularly Observed Long-Memory Lévy-Driven Moving Average Processes

Problem Statement
The paper investigates the asymptotic behavior of normalized partial sums for continuous-time moving average processes driven by a Lévy process, specifically when these processes are observed at random, irregular time points defined by a renewal process. The underlying continuous-time process, $X_t$ , is a long-memory process characterized by a kernel $a(t)$ decaying as $t^{d-1}$ with $0 < d < 1/2$ . The observation scheme involves sampling $X_t$ at epochs $T_k$ generated by a renewal process with inter-arrival times $\Delta_i$ .

The study addresses two distinct regimes for the renewal increments:

Finite-mean case (A1): $E(\Delta_1) < \infty$ .
Heavy-tailed case (A2): $P(\Delta_1 > x) = x^{-\alpha}\ell(x)$ with $0 < \alpha \le 1$ and infinite mean.

The core challenge lies in extending existing discrete-time results (specifically those by Ould Haye and Philippe, 2025) to the continuous-time setting. The authors note that while the limiting distributions in the discrete framework are known, the extension to continuous time is non-routine because key ingredients of the discrete proofs are unavailable in the continuous setting.

Methodology
The proof strategy is divided into two primary components: establishing asymptotic normality for the stochastic integral component and analyzing the random normalization factor.

Continuous-Time Lindeberg Proposition:
The authors first establish a sufficient condition for the asymptotic normality of stochastic integrals of the form $Z_n = \int_{-\infty}^{\infty} f_n(s) dL_s$ .
- Proposition 1 provides a continuous-time analogue of the Lindeberg theorem. It states that if the sequence of functions $f_n$ satisfies $\int f_n^2 \to 1$ and a specific "smallness" condition (the integral of $f_n^2$ over the set where $|f_n| > \eta$ vanishes), then the integral converges in distribution to a standard normal variable $N(0,1)$ .
- This is proven using the characteristic function of the Lévy process and Lemma 1, which expresses the characteristic function of the stochastic integral in terms of the Lévy exponent $\Psi$ .
Analysis of the Sampled Process:
The partial sums of the sampled process, $S_n = \sum_{i=1}^n X_{T_i}$ , are decomposed into a normalized stochastic integral and a random scaling factor $R_n$ .
- The authors define $f_n(s)$ such that $S_n = R_n \int f_n(s) dL_s$ .
- Step 1: They verify that the normalized integral $\int f_n(s) dL_s$ satisfies the conditions of Proposition 1, ensuring it converges to a standard normal distribution. This relies on Lemma 2, which proves the necessary truncation condition for the kernel $f_n$ .
- Step 2: They analyze the asymptotic behavior of the random normalization $R_n^2$ . This involves studying the variance of the partial sums and the interaction between the renewal process and the memory parameter $d$ .
- Step 3: They establish the asymptotic independence between the normalization factor $R_n$ and the stochastic integral in the specific regime where the limit is a variance mixture.

Key Contributions and Results
The main result, Theorem 1, characterizes the limiting distribution of the normalized partial sums $(Var(S_n))^{-1/2} S_n$ based on the interplay between the renewal tail index $\alpha$ and the memory parameter $d$ :

Gaussian Limit: If the renewal process has a finite mean ( $E(\Delta_1) < \infty$ ), or under heavy-tailed sampling where $\alpha = 1$ or $\alpha \le 1 - 2d$ , the normalized sums converge to a standard normal distribution $N$ .
Normal Variance Mixture: Under heavy-tailed sampling where $1 - 2d < \alpha < 1$ $1 - 2 d < α < 1$ , the limit is a normal variance mixture of the form $\sqrt{Z(\alpha, d)} N$ $Z (α, d) N$ .
- Here, $N$ is a standard normal variable independent of $Z(\alpha, d)$ .
- $Z(\alpha, d)$ is a random variable defined by a double integral involving a stable subordinator $B_t$ with parameter $\alpha$ :
  $Z(\alpha, d) = C_{\alpha, 1-2d} \int_{[0,1]^2} |B_x - B_y|^{2d-1} dx dy$
- The constant $C_{\alpha, r}$ is explicitly defined in terms of Gamma functions.

Technical Nuances
The paper highlights specific difficulties in the continuous-time extension:

In the discrete case, inter-arrival times are naturally bounded below by 1. In the continuous case, $\Delta_i > 0$ but can be arbitrarily small.
This distinction requires new arguments for the case $E(\Delta_1) < \infty$ and the boundary case $\alpha = 1 - 2d$ (specifically when the sequence $b_n^{-\alpha}$ is not summable).
To handle the boundary case, the authors decompose the variance terms using indicator functions based on the number of large jumps in the renewal process, utilizing Chernoff bounds to control the probability of small increments.

Significance and Claims
The authors claim that while the limiting distributions in this continuous-time setting coincide with those previously established for discrete-time processes, the mathematical derivation is significantly more complex. The paper successfully bridges the gap between discrete and continuous frameworks for irregularly sampled long-memory processes.

The work extends the theoretical understanding of how random time changes (renewal sampling) affect the dependence structure and asymptotic behavior of Lévy-driven processes. It confirms that the "destruction" of Gaussianity and the emergence of variance mixtures in the heavy-tailed regime are robust features that persist even when the underlying process is continuous and driven by general Lévy noise, not just discrete linear processes. The paper does not propose new applications but rather solidifies the theoretical foundation for modeling such irregularly observed phenomena in fields like finance and signal processing.

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