  We consider the first serial correlation coefficient under an AR(1) model
where errors are not assumed to be Gaussian. In this case it is necessary to
consider bootstrap approximations for tests based on the statistic since the
distribution of errors is unknown. We obtain saddle-point approximations for
tail probabilities of the statistic and its bootstrap version and use these to
show that the bootstrap tail probabilities approximate the true values with
given relative errors, thus extending the classical results of Daniels
[Biometrika 43 (1956) 169-185] for the Gaussian case. The methods require
conditioning on the set of odd numbered observations and suggest a conditional
bootstrap which we show has similar relative error properties.
