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This paper considers the problem of blind symbol rate estimation of signals linearly modulated by a sequence of unknown symbols. Oversampling the received signal generates cyclostationary statistics that are exploited to devise symbol-rate estimators by maximizing in the cyclic domain a (possibly weighted) sum of modulus squares of cyclic correlation estimates. Although quite natural, the asymptotic (large sample) performance of this estimator has not been studied rigorously. The consistency and asymptotic normality of this symbol-rate estimator is established when the number of samples N converges to infinity. It is shown that this estimator exhibits a fast convergence rate (proportional to N-32/), and it admits a simple closed-form expression for its asymptotic variance. This asymptotic expression enables performance analysis of the rate estimator as a function of the number of estimated cyclic correlation coefficients and the weighting matrix. A justification for the high performance of the unweighted estimator in high signal-to-noise scenarios is also provided.