Some Convergence For Nonstationary GARCH(1,1) Models

A GARCH(1,1) model is defined aswith initial values y0?0 and h0?0 a.s., where ?>0 ,a ?0 ,b?0 and {?,?t, t?1}is a sequence of independent and identically distributed random variables.When the model is nonstationary, that is Elogb(b+a?2)?0 , under suitable moment conditions or tailed probability conditions, the Marcinkiewicz-Zygmund strong (weak)law of large numbers and Hartmann-Winter law of the iterated logarithm, the generalized central limit theorem and the corresponding Chover law of the iterated logarithm are obtained. And the simulations are given for the Marcinkiewicz-Zygmund strong law of large numbers.