Exchange of limits: Why iterative decoding works

We consider communication over a family of binary-input memoryless output-symmetric channels using low-density parity-check codes under message passing decoding. The asymptotic (in the length) performance of such a combination for a fixed number of iterations is given by density evolution. It is customary to define the threshold of density evolution as the maximum channel parameter for which the bit error probability under density evolution converges to zero as a function of the iteration number. In practice we often work with short codes and perform a large number of iterations. It is therefore interesting to consider what happens if in the standard analysis we exchange the order in which the blocklength and the number of iterations diverge to infinity. In particular, we can ask whether both limits give the same threshold. Although empirical observations strongly suggest that the exchange of limits is valid for all channel parameters, we limit our discussion to channel parameters below the density evolution threshold. Specifically, we show that under some suitable technical conditions the bit error probability vanishes below the density evolution threshold regardless of how the limit is taken.