Compressive sensing (CS) technologies present many advantages over other existing approaches for implementing wideband spectrum sensing in cognitive radios (CRs), such as reduced sampling rate and computational complexity. However, there are two significant challenges: 1) choosing an appropriate number of sub-Nyquist measurements, and 2) deciding when to terminate the greedy recovery algorithm that reconstructs wideband spectrum. In this paper, an autonomous compressive spectrum sensing (ACSS) framework is presented that enables a CR to automatically choose the number of measurements while guaranteeing the wideband spectrum recovery with a small predictable recovery error. This is realized by the proposed measurement infrastructure and the validation technique. The proposed ACSS can find a good spectral estimate with high confidence by using only a small testing subset in both noiseless and noisy environments. Furthermore, a sparsity-aware spectral recovery algorithm is proposed to recover the wideband spectrum without requiring knowledge of the instantaneous spectral sparsity level. Such an algorithm bridges the gap between CS theory and practical spectrum sensing. Simulation results show that ACSS can not only recover the spectrum using an appropriate number of measurements, but can also considerably improve the spectral recovery performance compared with existing CS approaches. The proposed recovery algorithm can autonomously adopt a proper number of iterations, therefore solving the problems of under-fitting or over-fitting which commonly exist in most greedy recovery algorithms.