  This paper develops the limit theory of the GARCH(1,1) process that
moderately deviates from IGARCH process towards both stationary and explosive
regimes. The GARCH(1,1) process is defined by equations $u_t = \sigma_t
\varepsilon_t$, $\sigma_t^2 = \omega + \alpha_n u_{t-1}^2 +
\beta_n\sigma_{t-1}^2$ and $\alpha_n + \beta_n$ approaches to unity as sample
size goes to infinity. The asymptotic theory developed in this paper extends
Berkes et al. (2005) by allowing the parameters to have a slower convergence
rate. The results can be applied to unit root test for processes with
mildly-integrated GARCH innovations (e.g. Boswijk (2001), Cavaliere and Taylor
(2007, 2009)) and deriving limit theory of estimators for models involving
mildly-integrated GARCH processes (e.g. Jensen and Rahbek (2004), Francq and
Zako\"ian (2012, 2013)).
