  We introduce ellipticity criteria for random walks in i.i.d. random
environments under which we can extend the ballisticity conditions of
Sznitman's and the polynomial effective criteria of Berger, Drewitz and Ramirez
originally defined for uniformly elliptic random walks. We prove under them the
equivalence of Sznitman's (T') condition with the polynomial effective
criterion (P)_M, for M large enough. We furthermore give ellipticity criteria
under which a random walk satisfying the polynomial effective criterion, is
ballistic, satisfies the annealed central limit theorem or the quenched central
limit theorem.
