### abstract

- The spectral gap $\gamma$ of a finite, ergodic, and reversible Markov chain is an important parameter measuring the asymptotic rate of convergence. In applications, the transition matrix $P$ may be unknown, yet one sample of the chain up to a fixed time $n$ may be observed. We consider here the problem of estimating $\gamma$ from this data. Let $\pi$ be the stationary distribution of $P$, and $\pi_\star = \min_x \pi(x)$. We show that if $n = \tilde{O}\bigl(\frac{1}{\gamma \pi_\star}\bigr)$, then $\gamma$ can be estimated to within multiplicative constants with high probability. When $\pi$ is uniform on $d$ states, this matches (up to logarithmic correction) a lower bound of $\tilde{\Omega}\bigl(\frac{d}{\gamma}\bigr)$ steps required for precise estimation of $\gamma$. Moreover, we provide the first procedure for computing a fully data-dependent interval, from a single finite-length trajectory of the chain, that traps the mixing time $t_{\text{mix}}$ of the chain at a prescribed confidence level. The interval does not require the knowledge of any parameters of the chain. This stands in contrast to previous approaches, which either only provide point estimates, or require a reset mechanism, or additional prior knowledge. The interval is constructed around the relaxation time $t_{\text{relax}} = 1/\gamma$, which is strongly related to the mixing time, and the width of the interval converges to zero roughly at a $1/\sqrt{n}$ rate, where $n$ is the length of the sample path.