The square root rule for adaptive importance sampling

In adaptive importance sampling, and other contexts, we have unbiased and uncorrelated estimates of a common quantity $\mu$ and the variance of the $k$'th estimate is thought to decay like $k^{-y}$ for an unknown rate parameter $y\in [0,1]$. If we combine the estimates as though $y=1/2$, then the resulting estimate attains the optimal variance rate with a constant that is too large by at most $9/8$ for any $0\le y\le 1$ and any number $K$ of estimates.