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We study the asymptotic behavior of piecewise constant least squaresregression estimates, when the number of partitions of the estimate ispenalized. We show that the estimator is consistent in the relevant metric ifthe signal is in $L^2([0,1])$, the space of c\`{a}dl\`{a}g functions equippedwith the Skorokhod metric or $C([0,1])$ equipped with the supremum metric.Moreover, we consider the family of estimates under a varying smoothingparameter, also called scale space. We prove convergence of the empirical scalespace towards its deterministic target.

Scale space consistency of piecewise constant least squares estimators – another look at the regressogram