Regularising linear inverse problems under unknown non-Gaussian white noise allowing repeated measurements

We deal with the solution of a generic linear inverse problem in the Hilbert space setting. The exact right hand side is unknown and only accessible through discretised measurements corrupted by white noise with unknown arbitrary distribution. The measuring process can be repeated, which allows to reduce and estimate the measurement error through averaging. We show convergence against the true solution of the infinite-dimensional problem for a priori and a posteriori regularisation schemes as the number of measurements and the dimension of the discretisation tend to infinity under natural and easily verifiable conditions for the discretisation.