Limit theorems for Hilbert-valued random variables with Martingale perturbations

Abstract

This paper studies limit theorems for perturbed Hilbert-valued random variables under martingale-type errors. The observed sequence is modeled as an original sequence plus a martingale difference perturbation and an asymptotically negligible remainder term. We establish weak and strong laws of large numbers and central limit theorems under suitable moment and negligibility assumptions. In the negligible perturbation regime, the Gaussian limit of the original sequence is preserved. When the martingale perturbation contributes at the central limit scale, we obtain a Gaussian limit with a modified covariance operator. These results provide a perturbation framework for limit theorems in Hilbert spaces.