  Linear two-timescale stochastic approximation (SA) scheme is an important
class of algorithms which has become popular in reinforcement learning (RL),
particularly for the policy evaluation problem. Recently, a number of works
have been devoted to establishing the finite time analysis of the scheme,
especially under the Markovian (non-i.i.d.) noise settings that are ubiquitous
in practice. In this paper, we provide a finite-time analysis for linear two
timescale SA. Our bounds show that there is no discrepancy in the convergence
rate between Markovian and martingale noise, only the constants are affected by
the mixing time of the Markov chain. With an appropriate step size schedule,
the transient term in the expected error bound is $o(1/k^c)$ and the
steady-state term is ${\cal O}(1/k)$, where $c>1$ and $k$ is the iteration
number. Furthermore, we present an asymptotic expansion of the expected error
with a matching lower bound of $\Omega(1/k)$. A simple numerical experiment is
presented to support our theory.
