A generalization of a theorem by M. V. Tamhankar

Tamhankar [2] showed that, under suitable conditions, if X1, ..., Xn are independent random variables, then they are normally distributed with zero means and equal variances if and only if R is independent of ([Theta]1, ..., [Theta]n-1), R and [Theta]1, ..., [Theta]n-1 being the corresponding spherical coordinates. It is shown below that if (X1, ..., X8) and (X8+1, ..., Xn) are two independent random vectors having a continuous joint density function which is nonzero, then X1, ..., Xn are independent and normally distributed with zero means and equal variances if and only if for some integer l [set membership, variant] {1, ..., n-1}, (R, [Theta]1, ..., [Theta]l-1) and ([Theta]l, ..., [Theta]n-1) are independent.