**Working Papers, School of Business, Örebro University**
# No 2017:5:

Central limit theorems for functionals of large sample covariance matrix and mean vector in matrix-variate location mixture of normal distributions

*Taras Bodnar *()

*, Stepan Mazur *()

* and Nestor Parolya *()

**Abstract:** In this paper we consider the asymptotic distributions of
functionals of the sample covariance matrix and the sample mean vector
obtained under the assumption that the matrix of observations has a
matrix-variate location mixture of normal distributions. The central limit
theorem is derived for the product of the sample covariance matrix and the
sample mean vector. Moreover, we consider the product of the inverse sample
covariance matrix and the mean vector for which the central limit theorem
is established as well. All results are obtained under the
large-dimensional asymptotic regime where the dimension p and the sample
size n approach to in nity such that p=n ! c 2 [0;+1) when the sample
covariance matrix does not need to be invertible and p=n ! c 2 [0; 1)
otherwise.

**Keywords:** Normal mixtures; skew normal distribution; large dimensional asymptotics; stochastic representation; random matrix theory; (follow links to similar papers)

**JEL-Codes:** C00; C13; C15; (follow links to similar papers)

30 pages, August 22, 2017

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