For i.i.d. variables X1, X2,... , let An be the smallest σ -algebra for which X1,..., Xn are measurable and Bn the smallest σ -algebra for which Xn+1, Xn+2,... , are measurable. Let Sn be the σ -algebra of sets in An whose indicator functions are symmetric functions of X1,..., Xn . Let Cn be the σ -algebra generated by Sn and Bn . Show that for any sequence of U -statistics {Un, n ≥ m} generated as above by a random variable f (X1,..., Xm ) with f symmetric and EQ | f | ∞, {Un, Cn }n≥m is a reversed martingale.