The error exponent of zero-rate multiterminal hypothesis testing for sources with common information

We consider the multiterminal hypothesis testing problem with zero-rate constraint. For this problem, an upper bound on the optimal error exponent is given by Shalaby and Papamarcou, provided that the positivity condition holds. Our contribution is to prove that Shalaby and Papamarcou's upper bound is valid under a weaker condition: (i) two remote observations have a common random variable in the sense of Gácks and Körner, and (ii) when the value of the common random variable is fixed, the conditional distribution of remaining random variables satisfies the positivity condition.