Extrinsic Mean of Brownian Distributions on Compact Lie Groups

This paper studies Brownian distributions on compact Lie groups. These are defined as the marginal distributions of Brownian processes and are intended as a natural extension of the well-known normal distributions to compact Lie groups. It is shown that this definition preserves key properties of normal distributions. In particular, Brownian distributions transform in a nice way under group operations and satisfy an extension of the central limit theorem. Brownian distributions on a compact Lie group G belong to one of two parametric families N_L(g,C) and N_R(g,C)-g ∈ G and C a positive-definite symmetric matrix. In particular, the parameter g appears as a location parameter. An approach based on the extrinsic mean for estimation of the parameters g and C is studied in detail. It is shown that g is the unique extrinsic mean for a Brownian distribution N_L(g,C) or N_R(g,C). Resulting estimates are proved to be consistent and asymptotically normal. While they may also be used to simultaneously estimate g and C, it is seen this requires that G be embedded into a higher dimensional matrix Lie group. Going beyond Brownian distributions, it is shown the extrinsic mean can be used to recover the location parameter for a wider class of distributions arising more generally from Lévy processes. The compact Lie group structure places limitations on the analogy between normal distributions and Brownian distributions. This is illustrated by the study of multivariate Brownian distributions. These are introduced as Brownian distributions on some product group-e.g., G ×G. This paper describes their covariance structure and considers its transformation under group operations.