On E∏i=0kTr{Wmi}${\mathbb {E}}\left[\prod _{i=0}^k \operatorname{Tr}\lbrace W^{m_i}\rbrace\right]$, where W∼Wp(I,n)$W\sim \mathcal {W}_p(I,n)$

Jolanta Pielaszkiewicz, Dietrich Von Rosen and Martin Singull

Communications in Statistics - Theory and Methods, 2017, vol. 46, issue 6, 2990-3005

Abstract: In this paper, we give a general recursive formula for E[∏i=0kTr{Wmi}]${\mathbb {E}}[\prod _{i=0}^k \operatorname{Tr}\lbrace W^{m_i}\rbrace ]$, where W∼Wp(I,n)$W\sim \mathcal {W}_p(I,n)$ denotes a real Wishart matrix. Formulas for fixed n, p are presented as well as asymptotic versions when np→n,p→∞c$\frac{{n}}{p}\overset{n,p\rightarrow \infty }{\rightarrow }c$, i.e., when the so called Kolmogorov condition holds. Finally, we show application of the asymptotic moment relation when deriving moments for the Marchenko–Pastur distribution (free Poisson law). A numerical illustration using implementation of the main result is also performed.

Date: 2017
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DOI: 10.1080/03610926.2015.1053942

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