Global Positioning System (GPS) data are analysed from three permanent sites, two in the Western Alps (Grasse, France, and Zimmerwald, Switzerland) and the third in the Po Basin (Torino, Italy), for the 2.5-year period from 1 January 1996 to 1 July 1998. An analysis of the stochastic properties of the position time series reveals a significant amount of spatially and temporally correlated noise, which best fits a model combining white noise and flicker noise. The coloured noise is drastically reduced by spatially filtering the time series, suggesting that it is not due to site-specific effects such as monument motion, but rather to noise sources common to the three sites, such as reference frame, satellite orbit or Earth orientation errors. We find velocity uncertainties (95 per cent confidence interval) of 2.7 mm yr−1 at GRAS, 2.3 mm yr−1 at ZIMM, and 2.9 mm yr−1 at TORI. The residual velocity in the Eurasian reference frame is statistically greater than zero at ZIMM only, with 2.4 ± 2.3 mm yr−1 of motion in a NW direction. These uncertainties place an upper bound on the expected deformation in the Western Alps. Velocities relative to stable Eurasia do not exceed 3 mm yr−1, corresponding to a maximum strain rate of 0.03 μstrain yr−1 over the GRAS–TORI–ZIMM triangle.