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Variable-structure stochastic automata (VSSA) are considered which interact with an environment and which dynamically learn the optimal action that the environment offers. Like all VSSA the automata are fully defined by a set of action-probability updating rules. However, to minimize the requirements on the random-number generator used to implement the VSSA, and to increase the speed of convergence of the automation, the case in which the probability-updating functions can assume only a finite number of values. These values discretize the probability space (0, 1) and hence they are called discretized learning automata. The discretized automata are linear because the subintervals of (0, 1) are of equal length. The authors prove the following results: (a) two-action discretized linear reward-penalty automata are ergodic and epsilon -optimal in all environments whose minimum penalty probability is less than 0.5; (b) there exist discretized two-action linear reward-penalty automata that are ergodic and epsilon -optimal in all random environments, and (c) discretized two-action linear reward-penalty automata with artificially created absorbing barriers are epsilon -optimal in all random environments.< >