  We show theoretically how a correlation of multiple measurements on a qubit
undergoing pure dephasing can be expressed as environmental noise filtering.
The measurement of such correlations can be used for environmental noise
spectroscopy, and the family of noise filters achievable in such a setting is
broader than the one achievable with a standard approach, in which dynamical
decoupling sequences are used. We illustrate the advantages of this approach by
considering the case of noise spectrum with sharp features at very low
frequencies. We also show how appropriately chosen correlations of a few
measurements can detect the non-Gaussian character of certain environmental
noises, particularly the noise affecting the qubit at the so-called optimal
working point.
