The problem of distributed testing against independence with variable-length coding is considered when the average and not the maximum communication load is constrained as in previous works. The paper characterizes the optimum type-II error exponent of a single-sensor single-decision center system given a maximum type-I error probability when communication is either over a noise-free rate-R link or over a noisy discrete memoryless channel (DMC) with stop-feedback. Specifically, let E denote the maximum allowed type-I error probability. Then the optimum exponent of the system with a rate-R link under a constraint on the average communication load coincides with the optimum exponent of such a system with a rate R/(1-ε) link under a maximum communication load constraint. A strong converse thus does not hold under an average communication load constraint. A similar observation also holds for testing against independence over DMCs. With variable-length coding and stopfeedback and under an average communication load constraint, the optimum type-II error exponent over a DMC of capacity C equals the optimum exponent under fixed-length coding and a maximum communication load constraint when communication is over a DMC of capacity C/(1 - ε).
