Marcinkiewicz-type strong laws for partially exchangeable arrays

Let Xi, Yj, and Zij (i, j [set membership, variant] Z+) be independent families of i.i.d. random variables uniformly distributed on [0, 1], and let 0 < p < 2. We obtain conditions on a Borel function f defined on [0, 1]3 such that thus extending a result due to McConnell and Rieders when p = 1. For 1 < p < 2, and f satisfying the same conditions, we establish the existence of all moments less than p of the associated maximal function An application to degenerate U-statistics of degree two is given.