An O(n) algorithm for generating uniform random vectors in n-dimensional cones

Unbiased random vectors i.e. distributed uniformly in n-dimensional space, are widely applied and the computational cost of generating a vector increases only linearly with n. On the other hand, generating uniformly distributed random vectors in its subspaces typically involves the inefficiency of rejecting vectors falling outside, or re-weighting a non-uniformly distributed set of samples. Both approaches become severely ineffective as n increases. We present an efficient algorithm to generate uniformly distributed random directions in n-dimensional cones, to aid searching and sampling tasks in high dimensions.