On the randomness in learning

Given a random binary sequence X⁽ⁿ⁾ of random variables, X _t, t = 1, 2, . . . , n, for instance, one that is generated by a Markov source (teacher) of order k* (each state represented by k* bits). Assume that the probability of the event X_t = 1 is constant and denote it by β. Consider a learner which is based on a parametric model, for instance a Markov model of order k, who trains on a sequence x ^(m) which is randomly drawn by the teacher. Test the learner's performance by giving it a sequence x⁽ⁿ⁾ (generated by the teacher) and check its predictions on every bit of x⁽ⁿ⁾. An error occurs at time t if the learner's prediction Y_t differs from the true bit value X_t. Denote by ξ⁽ⁿ⁾ the sequence of errors where the error bit ξ_t at time t equals 1 or 0 according to whether the event of an error occurs or not, respectively. Consider the subsequence ξ^(v) of ξ⁽ⁿ⁾ which corresponds to the errors of predicting a 0, i.e., ξ^(v) consists of the bits of ξ⁽ⁿ⁾ only at times t such that Y_t = 0. In this paper we compute an estimate on the deviation of the frequency of 1s of ξ^(v) from β. The result shows that the level of randomness of ξ^(v) decreases relative to an increase in the complexity of the learner.