I am trying to prove that $$\sum_0^T \mathbf x_t \varepsilon_t \overset {p}{\rightarrow} 0 $$
In this case $\ \mathbf x_t \varepsilon_t $ is a martingale difference sequence, $\varepsilon_t$ is non-Gaussian but independent identically distributed with mean $0$ and variance $\sigma^2$, $T$ is the total number of observations in the model and $\mathbf x_t$ and $\varepsilon_t$ are independent.
I have found a complicated proof which uses mixingales and uniform integrebility. Is there another, simpler way to prove this result?