Higher order cumulants of random vectors, differential operators, and applications to statistical inference and time series. Can be downloaded from http://www.pstat.ucsb.edu/faculty/jammalam/html/research%20publication_files/DiffOpApplAMSV12.pdf

Abstract. This paper provides a unified and comprehensive approach that is useful in deriving expres-sions for higher order cumulants of random vectors. The use of this methodology is then illustrated in 3 diverse and novel contexts, namely: (i) in obtaining a lower bound (Bhattacharya bound) for the variance-covariance matrix of a vector of unbiased estimators where the density depends on several parameters, (ii) in studying the asymptotic theory of multivariable statistics when the population is not necessarily Gaussian and finally, (iii) in the study of multivariate nonlinear time series models and in obtaining higher order cumulant spectra. The approach depends on expanding the characteristic functions and cumulant generating functions in terms of the Kronecker products of differential operators. Using Tensor calculus, McCullagh and Speed have obtained similar results for cumulants of random vectors, but our objective here is to derive such expressions using only elementary calculus of several variables and also to highlight some important applications in statistics