  We consider the problem of sequential sampling from a finite number of
independent statistical populations to maximize the expected infinite horizon
average outcome per period, under a constraint that the expected average
sampling cost does not exceed an upper bound. The outcome distributions are not
known. We construct a class of consistent adaptive policies, under which the
average outcome converges with probability 1 to the true value under complete
information for all distributions with finite means. We also compare the rate
of convergence for various policies in this class using simulation.
