  High order cumulant tensors carry information about statistics of
non-normally distributed multivariate data. In this work we present a new
efficient algorithm for calculation of cumulants of arbitrary order in a
sliding window for data streams. We showed that this algorithms enables
speedups of cumulants updates compared to current algorithms. This algorithm
can be used for processing on-line high-frequency multivariate data and can
find applications in, e.g., on-line signal filtering and classification of data
streams.
  To present an application of this algorithm, we propose an estimator of
non-Gaussianity of a data stream based on the norms of high-order cumulant
tensors.
  We show how to detect the transition from Gaussian distributed data to
non-Gaussian ones in a~data stream. In order to achieve high implementation
efficiency of operations on super-symmetric tensors, such as cumulant tensors,
we employ the block structure to store and calculate only one hyper-pyramid
part of such tensors.
