Precise asymptotics in some strong limit theorems for multidimensionally indexed random variables

Consider Z+d (d[greater-or-equal, slanted]2)--the positive d-dimensional lattice points with partial ordering [less-than-or-equals, slant], let {Xk,k[set membership, variant]Z+d} be i.i.d. random variables with mean 0, and set Sn=[summation operator]k[less-than-or-equals, slant]nXk, n[set membership, variant]Z+d. We establish precise asymptotics for [summation operator]nnr/p-2P(Sn[greater-or-equal, slanted][var epsilon]n1/p), and for , (0[less-than-or-equals, slant][delta][less-than-or-equals, slant]1) as [var epsilon][downward right arrow]0, and for as .