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A binary detection problem of the Neymann-Pearson type, in which the probability density functions used are inaccurate versions of the true ones, are considered. The performance of the above suboptimal detection scheme as the number of observations increases is investigated. A necessary and sufficient condition is given for the exponential convergence to zero of the two error probabilities as the number of observations increases. The condition is in terms of an inequality between differences of asymptotic per sample informational divergence expressions.