Geodesic
On Characterizing the Multivariate Linear Exponential Distribution1
Canadian mathematical bulletin, Tome 12 (1969) no. 5, pp. 567-572

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If x and y are independent p component column vectors, and the conditional distribution of x, given x+y = z, is known, what can be said about the distributions of x and y? This problem has been solved by Seshadri (1966) in the particular case when the conditional distribution of x, given x+y = z, is multivariate normal. In fact Seshadri′s paper implicitly contains a characterization of the multivariate linear exponential distribution (1) where A(x) is a function of x not involving the p component column vector w of constant terms. 
Kabe, D. G. On Characterizing the Multivariate Linear Exponential Distribution1. Canadian mathematical bulletin, Tome 12 (1969) no. 5, pp. 567-572. doi: 10.4153/CMB-1969-073-2
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     title = {On {Characterizing} the {Multivariate} {Linear} {Exponential} {Distribution1}},
     journal = {Canadian mathematical bulletin},
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     year = {1969},
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     doi = {10.4153/CMB-1969-073-2},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1969-073-2/}
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