# Timeseries problem with law of large numbers

Let us have an AR(1) model with individual efect $$y_t = \alpha + \theta y_{t-1} + \varepsilon_t$$

with $$|\theta|<1$$ for stacionarity and $$\varepsilon_i$$ i.i.d. from distribution with mean $$0$$ and variance $$\sigma^2$$.

As $$\varepsilon_{t}$$ and $$y_{t-1}$$ are independent and $$\mathbb{E}\, \varepsilon_{t}=0$$ I would like to show that $$\frac{1}{T} \sum_{t=1}^T \varepsilon_{t}y_{t-1} \xrightarrow[T\rightarrow \infty]{P} 0.$$ For this I cannot use strong law of large numbers as individual terms are dependent (for example with t=5 and t=4 $$\varepsilon_5 y_4$$ and $$\varepsilon_4 y_3$$ are not independent).

Is there another way to show this?

$$\mathrm{var}\left[\frac{1}{T}\sum_{t=1}^T \epsilon_ty_t{-1} \right]=\frac{1}{T^2}\sum_{s,t=1}^T\mathrm{cov}\left[\epsilon_ty_{t-1},\epsilon_s y_{s-1} \right]$$
The covariances are going decrease exponentially in $$|t-s|$$ (for large $$|t-s|$$), so the sum will be $$O(T)$$ for any fixed $$\theta$$ and the scaling by $$T^2$$ makes it go to zero. Convergence in mean square implies convergence in probability by Chebyshev's inequality.