# Convergence of types in k-space

Research paper by Patrick Billingsley

Indexed on: 01 Jun '66Published on: 01 Jun '66Published in: Probability Theory and Related Fields

#### Abstract

If random variables Xn converge in distribution to a nondegenerate random variable X, and if transformations anXn+bn of them, with an>0, also converge in distribution to X, then an→1 and bn→0. Moreover, the group of transformations x→ax+b that preserve the distribution of a nondegenerate random variable consists either of the identity alone, or of the identity together with the reflection through some point. For proofs see [4], [5], or [6]. This paper gives the corresponding results for k-dimensional random vectors.