  Motivated by the study of asymptotic behaviour of the bandit problems, we
obtain several strategy-driven limit theorems including the law of large
numbers, the large deviation principle, and the central limit theorem.
Different from the classical limit theorems, we develop sampling
strategy-driven limit theorems that generate the maximum or minimum average
reward. The law of large numbers identifies all possible limits that are
achievable under various strategies. The large deviation principle provides the
maximum decay probabilities for deviations from the limiting domain. To
describe the fluctuations around averages, we obtain strategy-driven central
limit theorems under optimal strategies. The limits in these theorem are
identified explicitly, and depend heavily on the structure of the events or the
integrating functions and strategies. This demonstrates the key signature of
the learning structure. Our results can be used to estimate the maximal
(minimal) rewards, and to identify the conditions of avoiding the Parrondo's
paradox in the two-armed bandit problem. It also lays the theoretical
foundation for statistical inference in determining the arm that offers the
higher mean reward.
