  Markov parameters play a key role in system identification. There exists many
algorithms where these parameters are estimated using least-squares in a first,
pre-processing, step, including subspace identification and multi-step
least-squares algorithms, such as Weighted Null-Space Fitting. Recently, there
has been an increasing interest in non-asymptotic analysis of estimation
algorithms. In this contribution we identify the Markov parameters using
weighted least-squares and present non-asymptotic analysis for such estimator.
To cover both stable and unstable systems, multiple trajectories are collected.
We show that with the optimal weighting matrix, weighted least-squares gives a
tighter error bound than ordinary least-squares for the case of non-uniformly
distributed measurement errors. Moreover, as the optimal weighting matrix
depends on the system's true parameters, we introduce two methods to
consistently estimate the optimal weighting matrix, where the convergence rate
of these estimates is also provided. Numerical experiments demonstrate
improvements of weighted least-squares over ordinary least-squares in finite
sample settings.
