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Orthogonal frequency division multiplexing (OFDM) systems have been used extensively in wireless communications in recent years; thus, there is significant interest in analyzing the properties of the transmitted signal in such systems. In particular, a large amount of work has focused on analyzing the variation of the complex envelope of the transmitted signal and on designing methods to minimize this variation. In this paper, it is established that the complex envelope of a bandlimited uncoded OFDM signal converges weakly to a Gaussian random process as the number of subcarriers goes to infinity. This shows that the properties of the OFDM signal will asymptotically approach those of a Gaussian random process over any finite time interval. The convergence proof is then extended to two important cases, namely, coded OFDM systems and systems with an unequal power allocation across subcarriers.