A lower bound for expectation of a convex functional

Let [phi] be a symmetric convex function from n to . Under certain conditional symmetric conditions on the random variables X1,...,Xn, the inequality:E[[phi](X1,...,Xn)] [greater-or-equal, slanted] E[maxi [less-than-or-equals, slant] i [less-than-or-equals, slant] n[phi](0,...,0, Xi, 0,...,0)] is derived. Conditions under which the strict inequality holds are also obtained. Application to nonlinear autoregressive models and symmetrization of random variables are given.