  Estimation of third-order statistics relies on the availability of a huge
amount of data records, which can pose severe challenges on the data collecting
hardware in terms of considerable storage costs, overwhelming energy
consumption, and unaffordably high sampling rate especially when dealing with
high-dimensional data such as wideband signals. To overcome these challenges,
this paper focuses on the reconstruction of the third-order cumulants under the
compressive sensing framework. Specifically, this paper derives a transformed
linear system that directly connects the cross-cumulants of compressive
measurements to the desired third-order statistics. We provide sufficient
conditions for lossless third-order statistics reconstruction via solving
simple least-squares, along with the strongest achievable compression ratio. To
reduce the computational burden, we also propose an approach to recover
diagonal cumulant slices directly from compressive measurements, which is
useful when the cumulant slices are sufficient for the inference task at hand.
All the proposed techniques are tested via extensive simulations. The developed
joint sampling and reconstruction approach to third-order statistics estimation
is able to reduce the required sampling rates significantly by exploiting the
cumulant structure resulting from signal stationarity, even in the absence of
any sparsity constraints on the signal or cumulants.
