Decision-directed decoder for a binary symmetric channel with unknown crossover probability

We consider a suboptimum decision-directed block decoder for a binary symmetric channel that makes use of past decoding decisions to update its estimate of the channel's initially unknown crossover probability. The decoder has a threshold list decoding rule that uses the current estimated crossover probability. The estimate is updated by means of a stochastic approximation algorithm. It is shown to converge toward the true crossover probability with a bias that decreases exponentially with the code's block length, provided it never "runs away" toward zero after dropping below a certain critical value. The probability that this runaway phenomenon ever occurs is bounded by an expression that is exponentially decreasing in the code's block length and in the weight assigned to the initial estimate.