Doppler-based planet surveys have discovered numerous giant planets but are incomplete beyond several AU. At larger star–planet separations, direct planet detection through high-contrast imaging has proven successful, but this technique is sensitive only to young planets and characterization relies upon theoretical evolution models. Here we demonstrate that radial velocity measurements and high-contrast imaging can be combined to overcome these issues. The presence of widely separated companions can be deduced by identifying an acceleration (long-term trend) in the radial velocity of a star. By obtaining high spatial resolution follow-up imaging observations, we rule out scenarios in which such accelerations are caused by stellar binary companions with high statistical confidence. We report results from an analysis of Doppler measurements of a sample of 111 M-dwarf stars with a median of 29 radial velocity observations over a median time baseline of 11.8 yr. By targeting stars that exhibit a radial velocity acceleration ("trend") with adaptive optics imaging, we determine that 6.5% ± 3.0% of M-dwarf stars host one or more massive companions with 1 < m/M_J < 13 and 0 < a < 20 AU. These results are lower than analyses of the planet occurrence rate around higher-mass stars. We find the giant planet occurrence rate is described by a double power law in stellar mass M and metallicity F ≡ [Fe/H] such that f(M, F) = 0.039^+0.056)_(-0.028)M^(0.8)^(+1.1)_(-0.9) 10^(3.8±1.2). Our results are consistent with gravitational microlensing measurements of the planet occurrence rate; this study represents the first model-independent comparison with microlensing observations
