On Powers of Stieltjes Moment Sequences,I

For a Bernstein function f the sequence s_n=f(1)·...· f(n) is a Stieltjes moment sequence with the property that all powers s_n^c,c>0 are again Stieltjes moment sequences. We prove that is Stieltjes determinate for c≤ 2, but it can be indeterminate for c>2 as is shown by the moment sequence , corresponding to the Bernstein function f(s)=s. Nevertheless there always exists a unique product convolution semigroup such that ρ_c has moments . We apply the indeterminacy of for c>2 to prove that the distribution of the product of p independent identically distributed normal random variables is indeterminate if and only if p≥ 3