Rank-one power weakly mixing non-singular transformations

We show that Chacon's non-singular type IIIlambda transformation T-lambda, 0 < lambda less than or equal to 1, is power weakly mixing, i.e. for all se quences of non-zero integers {k(1),...,k(r)}, T-lambda(k1) x ... x T-lambda (kr) is ergodic. We then show that in infinite measure, this condition is n ot implied by infinite ergodic index (having all finite Cartesian products ergodic), and that infinite ergodic index does not imply 2-recurrence.