In this paper we define a class of MCMC algorithms, the generalized self regenerative chains (GSR), generalizing the SR chain of Sahu and Zhigljavski (2001), which contains rejection sampling as a special case. We show that this class contains members that are asymptotically more efficient and converge faster than the SR chains. We also consider generalizations of the Metropolis - Hastings independent chains or Metropolized independent sampling, and for some of these algorithms we are able to give the convergence rates and establish a lower bound for the asymptotic efficiency. All these MCMC algorithms use a proposal distribution that is independent of the current state. We discuss such algorithms generally. We are in particular interested in the number of times a given proposed value occurs consecutively as a state of the chain. We consider this number as a random integer weight that links these algorithms also to importance sampling. We show that for the generalizations of the SR and independent chains the expected values of these weights characterize the stationary distribution.