arXiv subjects

Paulo Henrique da Costa

Publications and source records attributed to Paulo Henrique da Costa.

3 recordsLinked to original sources

arXiv2026

Ergodicity bounds in the Sliced Wasserstein distance for Schur stable autoregressive processes

Explicit calculations in dimension one show for Schur stable autoregressive processes with standard Gaussian noise that the ergodic convergence in the Wasserstein-$2$ distance is essentially given by the sum of the mean, which decays exponentially, and the standard deviation, which decays with twice the speed. This paper starts by showing new upper and lower multivariate affine transport bounds for the Wasserstein-$r$ distance for $r$ greater and equal to $1$. These bounds allow to formulate a novel sufficient (non-Gaussian) ergodic interpolation condition for the mentioned mean-variance behavior to take place in case of more general Schur stable multivariate autoregressive processes. All ergodic estimates are non-asymptotic with completely explicit constants. The main applications are precise thermalization bounds for Schur stable $\mathsf{AR}(p)$ and $\mathsf{ARMA}(p,q)$ models in Wasserstein and Sliced Wasserstein distance. In the sequel we establish with the help of coupling techniques explicit upper and lower exponential bounds for more general multivariate Schur stable autoregressive processes. This includes parallel sampling and the convergence of the empiricial means. The utility of our results in particular for the Sliced Wasserstein distance are confirmed by multivariate numerical experiments.

math.PR↗

arXiv2025

Forward stochastic integration for adapted processes w.r.t. Riemann-Liouville fractional Brownian motion (Full version)

This paper provides the time-dependent $L^2$-martingale representation of the forward stochastic integral where the driving noise is the Riemann-Liouville fractional Brownian motion with parameter $\frac{1}{2} < H < 1$ and the integrand is a square-integrable adapted process. As a by-product, we obtain the exact $L^2$-isometry of the forward stochastic integrals based on suitable conditions on time-dependent martingale representations of adapted integrands combined with the Nelson's stochastic derivative of the underlying Gaussian driving noise.

math.PR↗

arXiv2022

Stochastic n-point D-bifurcations of stochastic Lévy flows and their complexity on finite spaces

This article refines the classical notion of a stochastic D-bifurcation to the respective family of n-point motions for homogeneous Markovian stochastic semiflows, such as stochastic Brownian flows of homeomorphisms, and their generalizations. This notion essentially detects at which level $k\leq n$ the support of the invariant measure of the k-point bifurcation has more than one connected component. Stochastic Brownian flows and their invariant measures which were shown by Kunita (1990) to be rigid, in the sense of being uniquely determined by the $1$-and $2$-point motions, and hence only stochastic n-point bifurcation of level $n=1$ or $n=2$ can occur. For general homogeneous stochastic Markov semiflows this turns out to be false. This article constructs minimal examples of where this rigidity is false in general on finite space and studies the complexity of the resulting n-point bifurcations.

math.PR↗