Convolution roots and embeddings of probability measures on lie groups

Abstract

We show that for a large class of connected Lie groups G, viz. from class C described below, given a probability measure μ on G and a natural number n, for any sequence {v_i} of th convolution roots of μ there exists a sequence {z_i} of _nth elements of G, centralising the support of μ , and such that {Z_iν _iZ_i^-1}is relatively compact; thus the set of roots is relatively compact 'modulo' the conjugation action of the centraliser of suppµ. 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].