Momentum Variance Reduction

Consider stochastic gradient descent with momentum,

with $0 \le \beta < 1$. The stochastic gradients $g_t$ are i.i.d. with $\mathbb{E}[g] = \mu$ and $\text{Var}[$.

In the steady-state limit $t \to \infty$, the variance of the momentum term converges to

For $\beta = 0.8$, compute $\text{Var}[m_\infty]$.