# Convolution roots and embedding of probability measures on Lie groups

Dani, S. G. and McCrudden, M. (2006) Convolution roots and embedding of probability measures on Lie groups. Advances in Mathematics. ISSN 0001-8708 (In Press)

We show that for a large class of connected Lie groups $G$, viz. from \emph{class} $\mathcal{C}$ described below, given a probability measure μ on $G$ and a natural number $n$, for any sequence $\{\nu_i\}$ of $n$th convolution roots of μ there exists a sequence $\{z_i\}$ of elements of G, centralising the support of μ, and such that $\{z_i \nu_i \z_i^{-1}\} is relatively compact; thus the set of roots is relatively compact ‘modulo’ the conjugation action of the centraliser of supp μ. We also analyse the dependence of the sequence$\{z_i\}$on$n\$. The results yield a simpler and more transparent proof of the embedding theorem for infinitely divisible probability measures on the Lie groups as above, proved in [S.G. Dani, M. McCrudden, Embeddability of infinitely divisible distributions on linear Lie groups, Invent. Math. 110 (1992) 237–261].