  Fan and Lv (2008) proposed the path-breaking theory of sure independence
screening (SIS) and an iterative algorithm (ISIS) to effectively reduce the
predictor dimension for further variable selection approaches. Fan et al.
(2009) extended ISIS to generalized linear models and introduced the Vanilla
ISIS (Van-ISIS) algorithm, allowing selected predictors to be screened out in
upcoming iterations. The success of SIS depends on its sure screening property,
which was obtained by Fan and Lv (2008) under the marginal correlation
assumption. However, despite wide applications of ISIS and Van-ISIS in various
scientific fields, their sure screening properties have not been proved during
the past decade. To fill this gap, we prove the sure screening properties of
three different types of iterative algorithms for linear models without relying
on the marginal correlation assumption, where ISIS and Van-ISIS can be regarded
as two special cases of them.
