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Asymptotic expansion of the matrix-variate Kummer-Dirichlet type I distribution. (English) Zbl 1269.15038

A set \(\{U_{k}\}_{k=1}^{n}\) of \(m\times m\) positive definite random matrices with \(\sum_{i=1}^{n} U_{i} < I_{m}\) is said to have a matrix variate Kummer-Dirichlet type I distribution with parameters \(\alpha_{1}, \dots , \alpha_{n}, \beta,\) and \(\Lambda\) if their joint probability density function is given by \[ \begin{split} K_{1} (\alpha_{1}, \dots , \alpha_{n}, \beta,\Lambda)\\ e^{\mathrm{trace} (-\Lambda \sum_{i=1}^{n} U_{i})} \prod_{i=1}^{n} \det (U_{i}) ^{\alpha_{i}- (m+1)/2} \det (I_{m}- \sum_{i=1}^{n} U_{i})^{\beta- (m+1)/2},\end{split} \] where \(\Lambda\) is a \(m\times m\) symmetric matrix, \(\alpha_{i} > (m-1)/2,\) \(i=1, \dots ,n\), \(\beta> (m-1)/2,\) and \(K_{1} (\alpha_{1}, \dots , \alpha_{n}, \beta,\Lambda)\) is the normalizing constant (cf. [A. K. Gupta, L. Cardeño and D. K. Nagar, J. Appl. Math. 1, No. 3, 117–139 (2001; Zbl 1040.62040)]).
For \(m=1\), the multivariate Kummer-beta density appears. The case \(n=1\) has been studied by D. K. Nagar and A. K. Gupta [J. Aust. Math. Soc. 73, No. 1, 11–25 (2002; Zbl 0999.62038)].
In this paper, an asymptotic expansion of the matrix variate Kummer-Dirichlet type I density is deduced. The authors use some results on the confluent hypergeometric function of the matrix argument \(_{1}F_{1} (\alpha,\gamma;X) = \sum _{k=0}^{\infty} \sum _{\kappa} \frac{\alpha_{\kappa}}{\gamma_{\kappa}} \frac{C_{\kappa}(X)}{\kappa!}\), \(\operatorname{Re} (\alpha) > (m-1)/2\) and \(\operatorname{Re}(\gamma-\alpha)> (m-1)/2\), where \(C_{\kappa}(X)\) is the zonal polynomial of the \(m\times m\) symmetric matrix \(X\) corresponding to the ordered partition \(\kappa= (k_{1}, \dots ,k_{m})\), \(k_{1}+ \dots +k_{m}= k\), \(k_{1}\geq \dots \geq k_{m} \) and the generalized hypergeometric coefficient \((a)_{\kappa}\) is defined by \((a)_{\kappa}= \prod_{i=1}^{m} (a- \frac{i-1}{2})_{k_{i}}.\)

MSC:

15B52 Random matrices (algebraic aspects)
33C70 Other hypergeometric functions and integrals in several variables
60B20 Random matrices (probabilistic aspects)
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References:

[1] Yasunori Fujikoshi, J. Sci. Hiroshima Univ. Ser. A-I Math 34 pp 73– (1970)
[2] Gupta A. K., Matrix Variate Distributions 104 (2000) · Zbl 0935.62064
[3] Gupta A. K., Random Oper. Stochastic Equations 11 (2) pp 101– (2003) · Zbl 1051.60014 · doi:10.1515/156939703322386878
[4] Gupta Arjun K., J. Appl. Math 1 (3) pp 117– (2001) · Zbl 1040.62040 · doi:10.1155/S1110757X0100701X
[5] James A. T., Ann. Math. Statist 35 pp 475– (1964) · Zbl 0121.36605 · doi:10.1214/aoms/1177703550
[6] Nagar D. K., J. Aust. Math. Soc 73 (1) pp 11– (2002) · Zbl 0999.62038 · doi:10.1017/S1446788700008442
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