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.