I'm proving the conditions for strict stationarity of GARCH(1,1)-process: $$X_t=\sigma_t Z_t\qquad \sigma_t^2 = \alpha_0 + \alpha_1 X_{t-1}^2 + \beta_1\sigma_{t-1}^2.$$ We can rewrite the process to a stochastic recurrence relation with $Y_t=\sigma_t^2$, $B_t=\alpha_0$ an $A_t=\alpha_1 Z_{t-1}^2 + \beta_1$: $$Y_t=B_t+A_tY_{t-1}.$$ If we iterate this equation $k$ times, we get $$Y_t=\prod_{i=0}^k A_{t-i} Y_{t-k-1}+B_t\bigg(1+\sum_{i=0}^k\prod_{j=0}^{i-1} A_{t-j}\bigg)$$ since $B_{t-i}=\alpha_0$ for all $i=0,1,...$. Now, I can easily see the argument with the strong law of large numbers been used on the second term, but how do I argue that the first term $$\prod_{i=0}^kA_{t-i} Y_{t-k-1} \rightarrow 0$$ almost surely for $k\rightarrow\infty$ and fixed $t\in\mathbb{Z}$?



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