Extremal Shot Noise Processes and Their Connection to Random Cutout Sets

Keskeiset käsitteet

Extremal shot noise processes are a class of Markov processes that exhibit a simple connection with random cutout sets, which are the sets of real numbers left uncovered by Poisson random covering intervals on the positive half-line. This connection provides a new proof of the characterization of random cutout sets.

Tiivistelmä

The paper studies the fundamental properties of extremal shot noise (ESN) processes, a class of sawtooth Markov processes. ESN processes are defined as the supremum of a Poisson point process with intensity λ × μ, where μ is a Borel measure on (0,∞) with finite tail.

The key insights are:

1. ESN processes are Markov processes with Feller property. Their finite-dimensional laws, semigroup, and stationary distribution are explicitly characterized.

2. The generator of ESN processes is studied, and the cores for the generator are identified.

3. The first passage times, transience, and recurrence of ESN processes are analyzed. A dichotomy is established based on the integrability of the tail of the measure μ.

4. The closure of the zero set of an ESN process is shown to coincide with the random cutout set associated with the underlying Poisson point process.

5. This connection is used to provide a new proof of the Fitzsimmons-Fristedt-Shepp Theorem, which characterizes the law of the random cutout set. Key properties of the random cutout set, such as the regenerative property and the fact that it is a perfect set, are also derived from this representation.

The paper also presents explicit examples of ESN processes, including self-similar ESN processes and ESN processes with specific stationary distributions.

Tilastot

The tail of the measure μ satisfies ¯μ(x) < ∞ for all x > 0.
The integral I = ∫∞1 exp(∫1s ¯μ(v)dv)ds determines whether the ESN process is recurrent (I = ∞) or transient (I < ∞).
The integral J = ∫10 exp(∫1s ¯μ(v)dv)ds determines whether the point 0 is accessible (J < ∞) or inaccessible (J = ∞) for the ESN process.

Lainaukset

"The closure of the zero set of the standard ESN(b, μ) process (M(t),t≥0), is of the following form {t≥0 : M(t) = 0} = [0,∞) - ∪s≥0(s, s + ξs/b) a.s.."
"When ∫∞1 ¯μ(y)dy < ∞, the process M admits a non-degenerate stationary distribution whose cumulative distribution function is given by π([0,u]) = exp(-1/b ∫∞u ¯μ(v)dv), for any u≥0."

Tärkeimmät oivallukset

Extremal shot noise processes and random cutout sets

by Clém... klo arxiv.org 10-01-2024

https://arxiv.org/pdf/2302.03082.pdf

Syvällisempiä Kysymyksiä

What are some potential applications of extremal shot noise processes beyond the study of random cutout sets?

Extremal shot noise processes (ESNs) have a wide range of applications beyond their connection to random cutout sets. One significant application is in the field of extreme value theory, where ESNs can model the behavior of maximum values in stochastic systems, particularly in spatial contexts. This is particularly relevant in environmental statistics, where extremes such as maximum rainfall or temperature can be modeled using ESNs to predict rare events.
Additionally, ESNs can be applied in finance, where they may be used to model the extremes of asset prices or returns, capturing the impact of sudden market shocks. The shot noise structure allows for the incorporation of time-dependent effects, which is crucial in financial modeling.
In telecommunications, ESNs can be utilized to model the noise in signal transmission, particularly in scenarios where the noise exhibits extreme behavior. This can help in designing more robust communication systems.
Moreover, ESNs can also find applications in queueing theory, where they can model the waiting times and service processes in systems with random arrivals and service times, particularly in scenarios where extreme delays are of interest.

How can the connection between ESN processes and random cutout sets be leveraged to study other properties of these random sets, such as their multifractal structure or their role in the study of Lévy processes?

The connection between ESN processes and random cutout sets provides a powerful framework for exploring various properties of these random sets. For instance, the multifractal structure of random cutout sets can be analyzed through the lens of ESNs by examining the scaling behavior of the zero sets of ESNs. The multifractal analysis can reveal how the distribution of gaps and clusters in the random cutout sets behaves at different scales, which is essential for understanding their geometric properties.
Furthermore, this connection can be exploited to study the zero sets of Lévy processes. Since ESNs are closely related to Lévy processes, particularly in their jump behavior, one can investigate how the properties of the random cutout sets reflect the underlying characteristics of the Lévy processes. For example, the distribution of the jump sizes and their impact on the coverage of the random cutout sets can provide insights into the long-term behavior of Lévy processes.
Additionally, the regenerative properties of random cutout sets, derived from their connection to ESNs, can be used to establish results regarding the recurrence and transience of Lévy processes, thereby enhancing our understanding of their long-term behavior.

Are there other classes of stochastic processes that exhibit similar connections to random geometric constructions, and how can such connections be exploited to gain new insights?

Yes, there are several classes of stochastic processes that exhibit connections to random geometric constructions. One notable example is random walks on random structures, such as random graphs or percolation clusters. The behavior of random walks on these structures can reveal insights into connectivity and the geometric properties of the underlying space.
Another class is branching processes, particularly those with random environments. The connection between branching processes and random geometric constructions can be leveraged to study the survival probabilities and extinction behaviors in heterogeneous environments, providing insights into population dynamics.
Gaussian processes also exhibit connections to random geometric constructions, particularly in the context of spatial statistics. The study of Gaussian random fields can lead to insights into the spatial distribution of phenomena, such as the clustering of points in a random field.
These connections can be exploited to gain new insights by applying techniques from potential theory, multifractal analysis, and geometric measure theory. For instance, understanding the zero sets of Gaussian processes can provide information about the topology and geometry of the underlying random fields, while the study of random walks on random structures can yield results about mixing times and convergence properties.
In summary, the interplay between stochastic processes and random geometric constructions opens up numerous avenues for research, allowing for a deeper understanding of complex systems across various fields.