OLS slope parameter variance in AR(1) and heteroskedastic errors


I am new here and I would like to ask for your help. I am writing thesis for bootstrapping financial time series and I need help with following issue.

AR1: X_t=alpha + beta*X_t-1 + e

As asymptotic variance of sqrt(n)(beta_hat-beta) converges to N(0,1-Beta^2) in homoskedastic case (errors in AR(1) have same variance), I can not figure out what is the asymptotic variance in conditional heteroskedastic case (ARCH(1)). I used to run some MonteCarlo simulations, and as the number of observation raises, the variance of sqrt(n)(Beta_hat-beta) raises too and it seems that there is no limit. Is that correct?

Estimates of Beta are computed with Ordinary Least Squares.
Don't you know some good literature about OLS asymptotics for autoregressions with ARCH/GARCH errors?