  Although complete randomization ensures covariate balance on average, the
chance for observing significant differences between treatment and control
covariate distributions increases with many covariates. Rerandomization
discards randomizations that do not satisfy a predetermined covariate balance
criterion, generally resulting in better covariate balance and more precise
estimates of causal effects. Previous theory has derived finite sample theory
for rerandomization under the assumptions of equal treatment group sizes,
Gaussian covariate and outcome distributions, or additive causal effects, but
not for the general sampling distribution of the difference-in-means estimator
for the average causal effect. To supplement existing results, we develop
asymptotic theory for rerandomization without these assumptions, which reveals
a non-Gaussian asymptotic distribution for this estimator, specifically a
linear combination of a Gaussian random variable and a truncated Gaussian
random variable. This distribution follows because rerandomization affects only
the projection of potential outcomes onto the covariate space but does not
affect the corresponding orthogonal residuals. We also demonstrate that,
compared to complete randomization, rerandomization reduces the asymptotic
sampling variances and quantile ranges of the difference-in-means estimator.
Moreover, our work allows the construction of accurate large-sample confidence
intervals for the average causal effect, thereby revealing further advantages
of rerandomization over complete randomization.
