Renewal-process approximations extend to Gaussian processes

What the study found: The authors report that some processes previously constructed from renewal processes can be used to approximate Gaussian processes, including fractional Brownian motion and multiple Stratonovich integrals. They also provide sufficient conditions on the renewal processes so that the convergence holds.

Why the authors say this matters: The study suggests that this renewal-process-based construction can be used beyond Brownian motion, including for fractional Brownian motion, which is a Gaussian process with any Hurst parameter. The authors present this as a way to build approximations of several Gaussian processes.

What the researchers tested: The paper builds on earlier work by Bardina and Rovira (2023), which constructed a family of processes from renewal processes that converge strongly toward Brownian motion. In this article, the authors examine whether some of those processes can also approximate other Gaussian processes and identify conditions on the renewal processes that support convergence.

What worked and what didn't: The abstract states that the approach works for approximating Gaussian processes such as fractional Brownian motion and multiple Stratonovich integrals. It also states that sufficient conditions are given to ensure convergence. The abstract does not describe any failed cases.

What to keep in mind: The available summary does not give the detailed conditions, proofs, or limits of the approximation result. It also does not describe any limitations beyond the scope of the examples mentioned.

Key points

• Some renewal-process-based processes can approximate Gaussian processes.
• The examples named in the abstract are fractional Brownian motion and multiple Stratonovich integrals.
• The authors give sufficient conditions on the renewal processes for convergence.
• The paper extends earlier work that converged strongly toward Brownian motion.
• Fractional Brownian motion is mentioned as an illustrative example with any Hurst parameter.