Adaptive importance sampling (AIS) constitutes new samples, such as particles in statistical physics, generated under certain probability distribution called policy \(q_k\) and the next policy \(q_{k+1}\) uses the new particles adaptively. In the earlier works, the policy is chosen as the kernel density estimate based on the previous particles reweighted by their importance weights. The authors propose a new approach called ‘safe adaptive importance sampling’ (SAIS) which estimates the policy as a mixture of kernel density estimate and certain ‘safe’ density with heavier tails. They also consider the functional approximation and derive convergence rates, leading to a central limit theorem for the estimates. It is observed that the asymptotic variance with this procedure is the same as that of an ‘oracle’ procedure. Further, a subsampling approach can be adopted to reduce the computational time involved without loosing the original efficiency. A simulation study at the end illustrates the practical nature of the algorithms developed. A section at the end gives detailed mathematical proofs including two appendices. There is a rich list of useful references.