Inference for Log-Lévy-Driven Cox Processes

Definitions and mathematical statement

A Cox process $N$ directed by a random locally finite measure $\Lambda$ is a point process which, conditional on $\Lambda$, is Poisson with intensity measure $\Lambda$. Thus counts on disjoint measurable sets are conditionally independent and $N(B)\mid\Lambda$ has Poisson distribution with mean $\Lambda(B)$.

In a conventional log-Lévy-driven construction on a Euclidean observation space, a Lévy random basis $L$ is an independently scattered infinitely divisible random measure, a field is formed as \[ Y_\theta(x)=\int k_\theta(x,s)\,L_\theta(ds),\qquad \Lambda_\theta(B)=\int_B e^{Y_\theta(x)}\,dx, \] and $N$ is the Cox process directed by $\Lambda_\theta$. This is explanatory notation, not a specification recovered from the source: existence of the stochastic integral and local finiteness of $\Lambda_\theta$ require appropriate assumptions, which are not supplied.

The source task is to develop statistical inference for log-Lévy-driven Cox processes, including their driving intensity and parameters. It does not choose an estimation target or observation regime. "Driving intensity" may mean the latent random intensity, its mean function, or characteristics of the underlying Lévy driver; these are distinct inferential targets. Likewise $\theta$ may encode the kernel and driver law, but no particular parameterization, finite-dimensionality or identifiability statement is given. Valid methods and their guarantees therefore require a specified model, observed data, target, regularity conditions and sampling regime. No estimator, likelihood approximation, rate or asymptotic distribution is imposed by the record.

The supplied triage checked 2026-08-21 labels this an open research direction. It says the model class and moment and mixing properties have been developed but general inference remains open for Lévy-driven and log-Lévy-driven Cox processes. The broader assessment is retained without conflating direct Lévy-driven intensities with the log-Lévy-driven construction asked for in the main statement. The registry also records manual review labeling the extracted research request exact and coherent.

Short English statement

Develop inference for a Cox point process with log-Lévy-driven intensity, covering its intensity and model parameters. The driver, kernel, observations and distinction between latent-intensity recovery and parameter estimation must be specified before a precise statistical theorem can be stated.

Sources

• Local UnsolvedMath registry, permanent ID 30001082, source code catalogue entry; supplied complete record and exact-statement manual-review note. Literature checked 2026-08-21: the assessment is supplied dated curation, not independently verified here.
• New Perspectives in Stochastic Geometry (2009), Oberwolfach Reports, DOI 10.4171/owr/2008/47, pp. 2655--2702: https://doi.org/10.4171/owr/2008/47. Proposed-year metadata gives 2008.
• Lévy-based Cox point processes: https://doi.org/10.1239/aap/1222868178.