  The paper investigates the problem of performing correlation analysis when
the number of observations is very large. In such a case, it is often necessary
to combine the random observations to achieve dimensionality reduction of the
problem. A novel class of statistical measures is obtained by approximating the
Taylor expansion of a general multivariate scalar function by a univariate
polynomial in the variable given as a simple sum of the original random
variables. The mean value of the polynomial is then a weighted sum of
statistical central sum-moments with the weights being application dependent.
Computing the sum-moments is computationally efficient and amenable to
mathematical analysis, provided that the distribution of the sum of random
variables can be obtained. Among several auxiliary results also obtained, the
first order sum-moments corresponding to sample means are used to reduce the
numerical complexity of linear regression by partitioning the data into
disjoint subsets. Illustrative examples are provided assuming the first and the
second order Markov processes.
