... the proof of Lemma 2 in ourproof of Theorem 1, we will include the full proof of Lemma 2 there.As examples of these lemmas, for the symmetric group, which has m = n(n − 1)/2roots, the asymptotic ... covariance form of X.LetYNbe the sum of N independent copies of X. For any vector v,letp(v) be the probability for the random walk, and let q(v) the density for the corresponding Brownianmotion ... stayswithin the Weyl chamber by integrating bκN(η, λ)overallλ in the Weyl chamber, and then relate this Brownian motion integral to the sum for random walks. the electronic journal of combinatorics...