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Random coding theorems and achievable rates for nonlinear additive noise channels are presented. Modeling the channel's nonlinear behavior as a causal, stationary Volterra system, upper bounds on the average error probability are obtained for maximum likelihood and weakly typical set decoding. The proposed bounds are deduced by treating correct decoding regions as subspaces of high concentration measure and deploying exponential martingale inequalities. Due to the union bound effect and the i.i.d. assumption imposed on the codewords components, the deduced exponents constitute only lower bounds on the true random coding exponents of nonlinear channels. Cubic and fourth-order nonlinearities are used as examples to illustrate the relation of the random coding exponents and achievable rates with respect to the channel's parameters.