## A form of multivariate gamma distribution.(English)Zbl 0760.62049

Summary: Let $$V_ i$$, $$i=1,\dots,k$$, be independent gamma random variables with shape $$\alpha_ i$$, scale $$\beta$$, and location parameter $$\gamma_ i$$, and consider the partial sums $$Z_ 1=V_ 1,Z_ 2=V_ 1+V_ 2,\dots,Z_ k=V_ 1+\dots+V_ k$$. When the scale parameters are all equal, each partial sum is again distributed as gamma, and hence the joint distribution of the partial sums may be called a multivariate gamma.
This distribution, whose marginals are positively correlated has several interesting properties and has potential applications in stochastic processes and reliability. We study this distribution as a multivariate extension of the three-parameter gamma and give several properties that relate to ratios and conditional distributions of partial sums. The general density, as well as special cases are considered.

### MSC:

 62H05 Characterization and structure theory for multivariate probability distributions; copulas 62H10 Multivariate distribution of statistics
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### References:

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