  Consider a countably infinite set of nodes, which sequentially make decisions
between two given hypotheses. Each node takes a measurement of the underlying
truth, observes the decisions from some immediate predecessors, and makes a
decision between the given hypotheses. We consider two classes of broadcast
failures: 1) each node broadcasts a decision to the other nodes, subject to
random erasure in the form of a binary erasure channel; 2) each node broadcasts
a randomly flipped decision to the other nodes in the form of a binary
symmetric channel. We are interested in whether there exists a decision
strategy consisting of a sequence of likelihood ratio tests such that the node
decisions converge in probability to the underlying truth. In both cases, we
show that if each node only learns from a bounded number of immediate
predecessors, then there does not exist a decision strategy such that the
decisions converge in probability to the underlying truth. However, in case 1,
we show that if each node learns from an unboundedly growing number of
predecessors, then the decisions converge in probability to the underlying
truth, even when the erasure probabilities converge to 1. We also derive the
convergence rate of the error probability. In case 2, we show that if each node
learns from all of its previous predecessors, then the decisions converge in
probability to the underlying truth when the flipping probabilities of the
binary symmetric channels are bounded away from 1/2. In the case where the
flipping probabilities converge to 1/2, we derive a necessary condition on the
convergence rate of the flipping probabilities such that the decisions still
converge to the underlying truth. We also explicitly characterize the
relationship between the convergence rate of the error probability and the
convergence rate of the flipping probabilities.
