Error Probability Bounds for Balanced Binary Relay Trees

We study the detection error probability associated with a balanced binary relay tree, where the leaves of the tree correspond to
N
identical and independent sensors. The root of the tree represents a fusion center that makes the overall detection decision. Each of the other nodes in the tree is a relay node that combines two binary messages to form a single output binary message. Only the leaves are sensors. In this way, the information from the sensors is aggregated into the fusion center via the relay nodes. In this context, we describe the evolution of the Type I and Type II error probabilities of the binary data as it propagates from the leaves toward the root. Tight upper and lower bounds for the total error probability at the fusion center as functions of
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are derived. These characterize how fast the total error probability converges to 0 with respect to
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, even if the individual sensors have error probabilities that converge to 1/2.