  Meta-analysis is a statistical method to combine results from multiple
clinical or genomic studies with the same or similar research problems. It has
been widely use to increase statistical power in finding clinical or genomic
differences among different groups. One major category of meta-analysis is
combining p-values from independent studies and the Fisher's method is one of
the most commonly used statistical methods. However, due to heterogeneity of
studies, particularly in the field of genomic research, with thousands of
features such as genes to assess, researches often desire to discover this
heterogeneous information when identify differentially expressed genomic
features. To address this problem, \citet{Li2011adaptively} proposed very
interesting statistical method, adaptively weighted (AW) Fisher's method, where
binary weights, $0$ or $1$, are assigned to the studies for each feature to
distinguish potential zero or none-zero effect sizes. \citet{Li2011adaptively}
has shown some good properties of AW fisher's method such as the admissibility.
In this paper, we further explore some asymptotic properties of AW-Fisher's
method including consistency of the adaptive weights and the asymptotic Bahadur
optimality of the test.
