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In, a new channel coding approach was proposed for random multiple access communication over the discrete-time memoryless channel. The coding approach allows users to choose their communication rates distributively without sharing the rate information with each other and the receiver. The receiver makes decoding and collision report decisions depending on whether reliable message recovery is possible. It was shown that, asymptotically as codeword length goes to infinity, the set of communication rates supporting reliable message recovery can be characterized by an achievable region which equals Shannon's information rate region without a convex hull operation. In this paper, we derive stronger versions of the coding theorems in by characterizing how system error probability, defined as the maximum of decoding error probability and collision miss detection probability, approaches zero with respect to the codeword length.