On Equivalence of Product Measure by Symmetrical Random l4-Translation

The converse of Sato and Watari′s theorem is proved. Let X = {Xn} be an i.i.d. random sequence with common density ƒ. If the distributions of X + Y and X are equivalent for every independent random sequence Y = {Yn} which is independent of X and Y ∈ l4 a.s., then it follows that ∫ (ƒ″)2/ƒ ≤ ∞