Let's say I have a sample of $X_1, \dots, X_n$, where I know that $X_i$ were generated by some ARCH(1) process. It means that $$X_i = \sigma_i z_i,$$ where $z_i \stackrel{iid}{\sim} N(0, 1)$ and $\sigma^2_i = \omega + \alpha X^2_{i-1}$.

I can estimate $z_i$ using following procedure:

1. Observing $X_1, \dots, X_n$, I can estimate $\hat{\omega}, \hat{\alpha}$ (MLE for example).
2. Then I can calculate $\hat{\sigma}^2_i$ using formula above with $\hat{\omega}, \hat{\alpha}$.
3. Finally I can estimate $\hat{z}_i = R_i / \hat{\sigma}_i$.

Problem: Obviously, when sample size grows, i.e. $n \to \infty$, then $(\hat{\omega}, \hat{\alpha}) \to (\omega, \alpha)$. Hence, $\hat{z}_i$ are "closer and closer" to real $z$ when $n \to \infty$. I am trying to formalize this statement, namely, I want to prove that: $$\sum^n_{i=1} P(\hat{z}_i \neq z_i) \to 0, \quad n \to \infty$$

I would be grateful for any support.

Let's say I have a sample of $X_1, \dots, X_n$, where I know that $X_i$ were generated by some ARCH(1) process. It means that $$X_i = \sigma_i z_i,$$ where $z_i \stackrel{iid}{\sim} N(0, 1)$ and $\sigma^2_i = \omega + \alpha X^2_{i-1}$.

I can estimate $z_i$ using following procedure:

1. Observing $X_1, \dots, X_n$, I can estimate $\hat{\omega}, \hat{\alpha}$ (MLE for example).
2. Then I can calculate $\hat{\sigma}^2_i$ using formula above with $\hat{\omega}, \hat{\alpha}$.
3. Finally I can estimate $\hat{z}_i = X_i / \hat{\sigma}_i$.

Problem: Obviously, when sample size grows, i.e. $n \to \infty$, then $(\hat{\omega}, \hat{\alpha}) \to (\omega, \alpha)$. Hence, $\hat{z}_i$ are "closer and closer" to real $z$ when $n \to \infty$. I am trying to formalize this statement, namely, I want to prove that: $$\sum^n_{i=1} P(\hat{z}_i \neq z_i) \to 0, \quad n \to \infty$$

I would be grateful for any support.