A generalization of a theorem by M. V. Tamhankar

AbstractTamhankar [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 (Θ1, …, Θn−1), R and Θ1, …, Θn−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 ∈ {1, …, n−1}, (R, Θ1, …, Θl−1) and (Θl, …, Θn−1) are independent