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Introduction. Consider a sequence of random variables {Xk:k = l,2, ■ • ■ ] obeying the law of large numbers, i.e., there exists a constant c such that for every e>0 the sequence of probabilities P{\n~12~2î-x^k-c\>e}=Pn(e) converges to zero as «-»<». The object of the present paper is to study the relationships among an exponential convergence rate (i.e., Pn(e) = 0(pn) for p=p(t) <1), the existence of the individual moment generating functions and the stochastic structure of the sequence \Xk\. Papers containing related studies (e.g., [l ; 3; 5]) have treated the case of independent random variables and demonstrated exponential convergence under the hypothesis that the moment generating functions exist.The present paper studies the extent to which an exponential convergence rate implies the existence of the moment generating function and conversely. In particular, satisfactory necessary and sufficient conditions (Theorem 2) are found for exponential convergence of sequences of independent (not necessarily identically distributed) random variables.In the first section it is proved that an exponential rate of convergence for any stationary sequence necessarily implies the existence of the moment generating function of the random variables. Conversely an example is constructed showing that restrictions on the size of the variables cannot be sufficient to insure exponential convergence for the general stationary sequence.