A simple condition for the boundedness of Sign-Perturbed-Sums (SPS) confidence regions

Sign-Perturbed-Sums (SPS) is a system identification algorithm that, under mild assumptions on the distribution of the noise, constructs confidence regions with finite-sample validity and a user-specified confidence level. For linear regression models, SPS regions are well-shaped in a precise meaning, but it is still possible (though rare in practice) that they are unbounded. In this communication, we provide a reformulation of a technical condition for the boundedness of the SPS regions in terms of a more practical excitation condition. We briefly argue that the simple condition here proposed provides insight to tune the SPS parameters, and even to design refined algorithms that can be guaranteed to deliver bounded regions.