Since the late 70’s, successive satellite missions have been monitoring the sun’s activity, recording the total solar irradiance (TSI). Some of these measurements last for more than a decade. It is then mandatory to merge them to obtain a seamless record whose duration exceeds that of the individual instruments. Climate models can be better validated using such long TSI records which can also help provide stronger constraints on past climate reconstructions (e.g.,back to the Maunder minimum). We propose a 3-stepmethod based on data fusion, including a stochastic noise model to take into account short and long-term correlations. Compared with previous products, the difference in terms of mean value over the whole time series and at the various solar minima are below 0.2W/m2. Next, we model the frequency spectrum of this 41-year TSI composite time series with a Generalized Gauss-Markov model to help describing an observed flattening at high frequencies. It allows us to fit a linear trend into these TSI time series by joint inversion with the stochastic noise model via a maximum-likelihood estimator. Our results show that the amplitude of such trend is∼−0.009±0.010 W/(m2.yr) for the period 1980-2021. These results are compared with the difference of irradiance values estimated from two consecutive solar minima. We conclude that the trend in these com-posite time series is mostly an artefact due to the coloured noise.