For \(\theta = (\theta_ 0, \theta_ 1, \theta_ 2)\) and \(x=(x_ 0,x_ 1,x_ 2)\) in \(R^ 3\), define \([\theta,x]\) as \(\theta_ 0 x_ 0 - \theta_ 1x_ 1 - \theta_ 2x_ 2\), \(C\) as \([x \in R^ 3:x_ 0>0\) and \([x,x]>0]\), \(R(x)\) as \(([x,x])^{1/2}\) and \(H_ 1\) as \([x \in C:x_ 0>0\), \(R(x)=1]\). Define the measure \(\sigma\) on \(H_ 1\) such that if \(\theta\) is in \(C\) and \(k=R(\theta)\), then \(\int \exp ([\theta,x]) \sigma (dx) = (k \exp k)^{-1}\). It is shown that there exists a positive measure \(\sigma_ \lambda\) in \(R^ 3\) such that its Laplace transform is \((k \exp k)^{-\lambda}\) if and only if \(\lambda \geq 1\). Then, for \(\lambda \geq 1\) and \(\theta\) in \(C\), defining \(P(\theta, \sigma_ \lambda) (dx)\) as \((k \exp k)^ \lambda \exp (-[\theta,x]) \sigma_ \lambda (dx)\), it is shown that if \(Y_ 0,\dots,Y_ n\) are independent random variables with density \(P(\theta, \sigma_{\lambda_ j})\), \(j=0,\dots,n\), and if \(S_ k = Y_ 0 + \cdots + Y_ k\) and \(Q_ k = R(S_ k)-R (S_{k-1}) - R(Y_ k)\), \(k=1,\dots,n\), then the \(n+1\) statistics \(D_ n = [\theta/k,S_ n] - R(S_ n)\), \(Q_ 1,\dots,Q_ n\) are independent random variables with the exponential \((k)\) or gamma \((1,1/k)\) distribution.