negative bias in the presence of high autocorrelation. Fortunately, a lugsail RBM estimator is easily implementable: $$\hat{\Sigma}_{R,L} = \dfrac{1}{1 - c} \hat{\Sigma}_{R,b} - \dfrac{c}{1 - c} \hat{\Sigma}_{R, \lfloor b/r \rfloor} \; ,$$

Our first main result, the proof for which is in Appendix \[subsec: strong\_consistency\_RBM\], demonstrates strong consistency of the lugsail RBM estimator. We will require the following theorem from [@kuel:phil:1980]. Let ${B\left(t\right)}$ be a $p$-dimensional standard Brownian motion.

Let Assumption \[ass:mixing\] hold. Then there exists a $p \times p$ lower triangular matrix $L$ such that $LL^T = \Sigma$, a non-negative increasing function $\psi(n) = n^{1/2 - \lambda}$ for some $\lambda > 0$, a finite random variable D, and a sufficiently rich probability space $\Omega$ such that for almost all $w \in \Omega$ such that for all $n > n_{0}$, with probability 1, $$\left\|\sum_{t=1}^{n}Y_{k,t} - n \mu - LB\left(n\right)\right\| < D(w)\psi(n) \;.$$

\[thm:RBM\_consistency\] If Assumption \[ass:mixing\] holds, $\hat{\Sigma}_{k,b} \overset{a.s.}{\to} \Sigma$ and $\left(b\log \log n\right)/n \to 0$, then $\hat{\Sigma}_{R,L}\overset{a.s.}{\to} \Sigma$ as $n \to \infty$.

The assumption, $\left(b\log \log n\right)/n \to 0$ as $n \to \infty$ is already required for the strong consistency of $\hat{\Sigma}_{k,b}$ [@dame:1994; @vats:fleg:jon:2019] and is not an added burden here.

Theorem \[thm:RBM\_consistency\] guarantees that sequential stopping rules of 