  A new class of stochastic processes called independent and periodically
identically distributed (i.p.i.d.) processes is defined to capture periodically
varying statistical behavior. A novel Bayesian theory is developed for
detecting a change in the distribution of an i.p.i.d. process. It is shown that
the Bayesian change point problem can be expressed as a problem of optimal
control of a Markov decision process (MDP) with periodic transition and cost
structures. Optimal control theory is developed for periodic MDPs for
discounted and undiscounted total cost criteria. A fixed-point equation is
obtained that is satisfied by the optimal cost function. It is shown that the
optimal policy for the MDP is nonstationary but periodic in nature. A value
iteration algorithm is obtained to compute the optimal cost function. The
results from the MDP theory are then applied to detect changes in i.p.i.d.
processes. It is shown that while the optimal change point algorithm is a
stopping rule based on a periodic sequence of thresholds, a single-threshold
policy is asymptotically optimal, as the probability of false alarm goes to
zero. Numerical results are provided to demonstrate that the asymptotically
optimal policy is not strictly optimal.
