Weak approximation for Gaussian processes from renewal processes
Xavier Bardina, Salim Boukfal, Marc Cano, Carles Rovira
TL;DR
The paper addresses weakly approximating general Gaussian processes, including fractional Brownian motion, using renewal-process-based constructions. It defines deterministic kernels $K$ and renewal-derived signals $\theta_n$, producing processes $Y_t^n=\int_0^1 K(t,x)\theta_n(x)dx$ that converge weakly to Gaussian processes with covariance $\int_0^1 K(t,x)K(s,x)dx$, and further shows convergence to vectors of iterated Stratonovich integrals for product-form test functions. The main contributions are twofold: (i) a weak-convergence result for Gaussian processes represented as Wiener-type integrals with a deterministic kernel, and (ii) a weak-convergence result to multiple Stratonovich integrals, all under mild interarrival hazard-rate conditions (G). This framework includes fractional Brownian motion as a special case and broadens the scope of renewal-based approximations beyond Brownian motion, offering new tools for simulation and analysis when Poisson-exponential assumptions are not available.
Abstract
In previous works, Bardina and Rovira (2023) constructed a family of processes that converge strongly towards Brownian motion, defined from renewal processes, are constructed. In this paper we prove that some of these processes can be utilized to build approximations of Gaussian processes such as fractional Brownian motion or multiple Stratonovich integrals and we provide sufficient conditions on renewal processes to ensure that the convergence holds. An illustrative example of such a Gaussian process is the fractional Brownian motion with any Hurst parameter.